Theory of cavity-enhanced spontaneous four wave mixing

Let us now turn to the case where the SFWM fiber is in the form of a cavity. This could be done in one of two ways: (i) two Bragg mirrors with a separation L could be written on a length of fiber thus forming a cavity, or (ii) a fiber or waveguide ring cavity, of circulation length L_c could be used, coupled to a straight fiber or waveguide placed tangentially to the ring. In this paper we will study cases where the cavity used is resonant to one or both of the SFWM modes, while it is not resonant to the pump frequency. Figure 1 shows, schematically, both of these types of fiber cavity sources. While we are interested in both the pulsed and continuous-wave pump regimes, we will carry out our analysis for the pulsed case and will obtain the continuous-wave case as an appropriate limit.

For simplicity, in this paper we restrict our attention to the SFWM process with degenerate pumps, i.e. for which both pump photons in a given event come from the same pump mode. In our analysis below, we will assume a cavity formed by two mirrors, although we will indicate how our results would apply to the case of a ring cavity. Because in our calculation the cavity is not resonant to the pump, as mentioned above, both mirrors are perfectly transmissive for the pump. We will refer to a fiber cavity SFWM source resonant only to the signal-photon as the Cs cavity, to a source resonant only to the idler-photon as a Ci cavity, and to a source resonant to both SFWM photons as a Csi cavity. We also assume that the left-hand cavity mirror (through which the pump enters, to be referred to as mirror 1) is perfectly reflective for SFWM photons, so that all emitted photons are forward-propagating, transmitted by the right-hand cavity mirror (to be referred to as mirror 2). The calculation below closely follows that for SPDC reported by us in an earlier paper, see [22].

For the Csi cavity it can be shown that the two-photon component of the state at the cavity output becomes

$$\begin{eqnarray} &&\vert {\Psi }_{2}^{s i}\rangle =\zeta \mathop{\sum }\limits _{{k}_{\mathrm{s}}}\mathop{\sum }\limits _{{k}_{\mathrm{i}}}{G}_{\mathrm{si}}({k}_{\mathrm{s}},{k}_{\mathrm{i}})a_{\mathrm{s}}^{\dagger }({k}_{\mathrm{ s}})a_{\mathrm{i}}^{\dagger }({k}_{\mathrm{ i}})\vert 0\rangle _{\mathrm{s}}\vert 0\rangle _{\mathrm{i}}, \end{eqnarray} \tag{ 5 }$$

where we have defined the joint amplitude function

$$\begin{eqnarray} &&{G}_{\mathrm{si}}({\omega }_{\mathrm{s}},{\omega }_{\mathrm{i}})=G[{k}_{\mathrm{s}}({\omega }_{\mathrm{s}}),{k}_{\mathrm{i}}({\omega }_{\mathrm{i}})]{A}_{\mathrm{s}}({\omega }_{\mathrm{s}}){A}_{\mathrm{i}}({\omega }_{\mathrm{i}}), \end{eqnarray} \tag{ 6 }$$

given in terms of G(k_s,k_i) (see equation (2)), but expressed here in terms of frequencies, and where the function A_μ(ω) describes the effect of the cavity and can be written as

$$\begin{eqnarray} &&{A}_{\mu }({\omega }_{\mu })=\frac{{t}_{2 \mu }}{1-\vert {r}_{2 \mu }\vert {\mathrm{e}}^{\mathrm{i}(2{\beta }_{\mu }+{\delta }_{1 \mu }+{\delta }_{2 \mu })}}. \end{eqnarray} \tag{ 7 }$$

In equation (7), t_2μ and r_2μ = |r_2μ|e^iδ_2μ are the amplitude transmissivity and the amplitude reflectivity of the right-hand cavity mirror, respectively, while r_1μ = e^iδ_1μ is the amplitude reflectivity of the left-hand cavity mirror, for μ = s,i. In equation (7), β_μ = k(ω_μ)L represents the phase acquired by photon μ in one round trip of the fiber cavity.

The cavity-modified JSA function is shown in equation (6). The corresponding cavity-modified joint spectral intensity (JSI) S_si(ω_s,ω_i) = |G_si(ω_s,ω_i)|² is given by [22]

$$\begin{eqnarray} &&{S}_{\mathrm{si}}({\omega }_{\mathrm{s}},{\omega }_{\mathrm{i}})=\vert F({\omega }_{\mathrm{s}},{\omega }_{\mathrm{i}})\vert ^{2}{\mathscr{A}}_{\mathrm{ s}}({\omega }_{\mathrm{s}}){\mathscr{A}}_{\mathrm{i}}({\omega }_{\mathrm{i}}), \end{eqnarray} \tag{ 8 }$$

with F(ω_s,ω_i) given by equation (2), and

$$\begin{eqnarray} &&{\mathscr{A}}_{\mu }({\omega }_{\mu })=\frac{\vert {t}_{2 \mu }\vert ^{2}}{\left (1-\vert {r}_{2 \mu }\vert \right )^{2}}\frac{1}{1+{\mathscr{F}}_{\mu }{\sin }^{2}[{\Delta }_{\mu }({\omega }_{\mu })/2]}, \end{eqnarray} \tag{ 9 }$$

written in terms of the coefficient of finesse _μ

$$\begin{eqnarray} &&{\mathscr{F}}_{\mu }=\frac{4\vert {r}_{2 \mu }\vert }{\left (1-\vert {r}_{2 \mu }\vert \right )^{2}}, \end{eqnarray} \tag{ 10 }$$

and the phase factor

$$\begin{eqnarray} &&{\Delta }_{\mu }({\omega }_{\mu })=2 \beta +{\delta }_{1 \mu }+{\delta }_{2 \mu }. \end{eqnarray} \tag{ 11 }$$

Note that in the case of a ring cavity mirror 1 does not exist, while mirror 2 becomes a fiber-based beam splitter as shown in figure 1(b). The coefficient r₂ is then the amplitude associated with an intra-cavity photon remaining in the cavity in a given iteration, while the coefficient t₂ is the amplitude associated with an intra-cavity photon exiting the cavity in a given iteration. Because mirror 1 does not exist, δ_1μ should be replaced with 0. Also, the fiber length L should be replaced with L_r/2, where L_r is the circulation length in the ring cavity.

The Airy function _μ(ω_μ) is formed by a sequence of equal height peaks, with width δω_μ, and spectral separation, or free spectral range, Δω_μ. Under the assumption that the index of refraction remains constant among different peaks, it can shown that [22]

$$\begin{eqnarray} &&\Delta {\omega }_{\mu }=\frac{\pi c}{L{n}_{\mu }}, \end{eqnarray} \tag{ 12 }$$

and

$$\begin{eqnarray} &&\delta {\omega }_{\mu }=\frac{2 c}{L{n}_{\mu }\sqrt{{\mathscr{F}}_{\mu }}}=\frac{2 \Delta \omega }{\pi \sqrt{{\mathscr{F}}_{\mu }}}, \end{eqnarray} \tag{ 13 }$$

where n_μ = n(ω_μ0) is the refractive index of the fiber evaluated at the central frequency of the signal and idler modes (μ = s,i). Note that the cavity is resonant at frequencies at which the peaks of the Airy function appear. Thus, in order to ensure resonance at a particular frequency ω, the condition sin[Δ_μ(ω)/2] = 0 must be fulfilled. Also note that resonance at a given desired frequency can be achieved by adjusting the reflection phases δ_1μ and δ_2μ.

For the case of a cavity resonant at only one of the SFWM modes, say the signal mode, the two-photon state and the JSI are given by equations (5) and (8), respectively, with |r_2i| = 0. Thus, _i = 0 and _i(ω_i) = 1, and consequently the JSI for the Cs cavity becomes

$$\begin{eqnarray} &&{S}_{\mathrm{s}}({\omega }_{\mathrm{s}},{\omega }_{\mathrm{i}})=\vert F({\omega }_{\mathrm{s}},{\omega }_{\mathrm{i}})\vert ^{2}{\mathscr{A}}_{\mathrm{ s}}({\omega }_{\mathrm{s}}). \end{eqnarray} \tag{ 14 }$$

Likewise, for a Ci cavity the JSI becomes

$$\begin{eqnarray} &&{S}_{\mathrm{i}}({\omega }_{\mathrm{s}},{\omega }_{\mathrm{i}})=\vert F({\omega }_{\mathrm{s}},{\omega }_{\mathrm{i}})\vert ^{2}{\mathscr{A}}_{\mathrm{ i}}({\omega }_{\mathrm{i}}). \end{eqnarray} \tag{ 15 }$$

The spectral structure of a SFWM two-photon state obtained with a Csi cavity is illustrated in figures 2(a)–(c), for which we have assumed a photonic crystal fiber (PCF) with 0.68 μm core radius and 0.5 air-filling fraction, and we have relied on the step-index model [37]. These parameters have been chosen to achieve phasematching for a SFWM process with degenerate pumps centered at ω_p = 2πc/1.064 μm and signal and idler modes centered at ${\omega }_{\mathrm{s}}^{o}=2 \pi c/0.8 5 2~\mu \mathrm{m}$ and ${\omega }_{\mathrm{i}}^{o}=2 \pi c/1.4 1 7~\mu \mathrm{m}$, respectively. The SFWM frequencies are chosen to be considerably frequency non-degenerate, which facilitates photon pair splitting. We have assumed a fiber length of L = 1 cm and a pump bandwidth of σ = 80 GHz. Figure 2(a) represents the JSI without a cavity, i.e. so that _s(ω) = _i(ω) = 1. The two-dimensional Airy pattern describing the effect of the cavity is shown in figure 2(b), while figure 2(c) represents the resulting JSI for the cavity-modified SFWM photon pairs. For illustrative purposes we have assumed here r₂ = 0.8, corresponding to  = 80, in a realistic implementation the finesse would probably be chosen much higher to ensure the emission of sufficiently narrowband photon pairs. Note that for these plots (figures 2(a)–(c)), we have assumed that the signal and idler photons are filtered using rectangular filters of width σ_fμ = 5Δω_μ (with μ = s,i), centered at ${\omega }_{\mathrm{s}}^{o}$ and ${\omega }_{\mathrm{i}}^{o}$, respectively.

Zoom In Zoom Out Reset image size

As can be noted, the effect of the cavity is that the emission frequencies are re-distributed, with respect to the corresponding emission frequencies in the absence of a cavity, so that emission occurs only for frequencies within the cavity modes. Because the mode width δω scales as ^−1/2, cavity-enhanced SFWM with a sufficiently high finesse can be an effective route towards narrowband photon pairs, to be used for example in the context of atom–photon interfaces. As will be studied further in a later section, if the pump bandwidth is of the order of the cavity mode width δω, a significant enhancement of the source brightness can result with respect to an equivalent source without a cavity.

In figures 2(d)–(f) we illustrate the spectral structure of SFWM photon pairs produced by a Cs cavity. Here we assume an identical source configuration to that assumed for the Csi cavity, except that the resonance for the idler photon has been suppressed. figure 2(d) represents the JSI without a cavity, i.e. so that _s(ω) = _i(ω) = 1. The one-dimensional Airy pattern describing the effect of a cavity resonant for the single mode only is shown in figure 2(e), while figure 2(f) represents the resulting JSI for the cavity-modified SFWM photon pairs. As for the Csi cavity, we have assumed that the signal and idler photons are filtered using rectangular filters with width σ_fμ = 5Δω_μ, centered at ${\omega }_{\mathrm{s}}^{o}$ and ${\omega }_{\mathrm{i}}^{o}$, respectively.

As may be appreciated, the effect of the cavity is a spectral re-distribution for the resonant, in this case signal, photon. The result is that the JSI is now formed by elongated modes of width δω for the signal photon and a larger width determined by the cavity length and pump bandwidth for the idler photon.

It is instructive to study the two-photon state in the temporal domain, in addition to the spectral domain used in our description so far. To this end, we define the joint temporal amplitude (JTA) $\tilde {f}({t}_{\mathrm{s}},{t}_{\mathrm{i}})$ as the two-dimensional Fourier transform of the JSA, i.e. 

$$\begin{eqnarray} &&\tilde {f}({t}_{\mathrm{s}},{t}_{\mathrm{i}})=\mathcal{F}\{ {G}_{\mathrm{si}}({\omega }_{\mathrm{s}},{\omega }_{\mathrm{i}})\} , \end{eqnarray} \tag{ 16 }$$

where denotes the Fourier transform and G_si(ω_s,ω_i) is given according equation (6).

The joint temporal intensity (JTI) can now be written as ${S}_{\mathrm{t}}({t}_{\mathrm{s}},{t}_{\mathrm{i}})=\vert \tilde {f}({t}_{\mathrm{s}},{t}_{\mathrm{i}})\vert ^{2}$, which when properly normalized yields the time of emission joint probability distribution.

The spectral cavity mode structure translates into a specific temporal-mode structure. In figure 3, we have shown for the same source parameters as in figure 2 the resulting JTI calculated numerically through equation (16). In physical terms, the two photons in a given pair may exit the cavity together, or may be delayed with respect to each other by a whole number of cavity round trip times. The vertical (horizontal) dotted lines indicate possible signal-mode (idler-mode) emission times, corresponding to a whole number of cavity round trip times. This results in a matrix of temporal emission modes, as is apparent in figure 3, which decay in amplitude for increasing times of emission (which imply an increasing number of cavity round trips). Of course, for a higher finesse, this amplitude roll-off is slower, resulting in the appearance of a larger number of temporal modes.

Zoom In Zoom Out Reset image size

Let us label each of the emission modes according to the cavity iteration at which each of the two photons is emitted, so that the state |ij〉, with a corresponding probability amplitude C_ij, means the signal photon emitted in the ith cavity iteration and the idler photon emitted in the jth cavity iteration. We may then write down the resulting two-photon state as follows:

$$\begin{eqnarray} \vert \Psi \rangle &=&[{C}_{0 0}\vert 0 0\rangle ]+[{C}_{1 0}\vert 1 0\rangle +{C}_{0 1}\vert 0 1\rangle ]+[{C}_{2 0}\vert 2 0\rangle \nonumber\\ &&+~{C}_{1 1}\vert 1 1\rangle +{C}_{0 2}\vert 0 2\rangle ]+[{C}_{3 0}\vert 3 0\rangle +{C}_{2 1}\vert 2 1\rangle \nonumber\\ &&+~{C}_{1 2}\vert 1 2\rangle +{C}_{0 3}\vert 0 3\rangle ]+\cdots . \end{eqnarray} \tag{ 17 }$$

Here, each group of terms in square brackets indicates a certain fixed number of total cavity iterations. Thus, the only term in the first square bracket corresponds to both photons emitted together without any intra-cavity reflections. The two terms in the second square bracket correspond to a total number of one cavity iterations between the two photons. The three terms in the third square-bracket correspond to a total number of two cavity iterations between the two photons, and so on for higher-order terms. It can be appreciated from figure 3(a) that all modes within a given square-bracket group involve the same amplitude.

Note that the state in equation (17) exhibits quantum entanglement in the temporal-mode degree of freedom. Note also that the state of each photon is described by a Hilbert space with a dimension of the order of the number of effective cavity round trip times before the state is extinguished. Thus, for a higher coefficient of finesse, there will be more cavity round trip times, and the dimension of the Hilbert space which describes each photon will increase. The triangular shape of the overall pattern of modes (see figure 3(a)) implies that the state is non-factorable. Thus, interestingly, this physical system leads to higher-dimensional entanglement in the temporal-mode degree of freedom with a dimension which can be tailored by adjusting the cavity finesse.

We can carry out a similar analysis for the Cs cavity. The resulting JTI is plotted in figure 3(b). The fact that only one photon is now resonant, implies that the temporal-mode structure now appears only for the signal photon. Note that this state may now be written as |Ψ〉 = C₀₀|00〉 + C₀₁|01〉 + C₀₂|02〉 + ⋯, which is factorable; hence, this system does not lead to entanglement in the temporal-mode degree of freedom.

In order to gain further physical insight into the nature of the two-photon state in the temporal domain, in figure 4(a) we plot the JTI for the same Csi cavity source, this time as a function of the time-sum t_s + t_i and time-difference t_s − t_i variables. Figure 4(b) is similar to the previous plot, except that the state has been spectrally filtered so that only the central spectral mode in the JSI is retained. It may be seen from these plots that the triangular pattern which appears in the latter JTI may be thought of as an "envelope" for the JTI corresponding to the full, i.e. spectrally unfiltered, state. It is interesting to consider the time of arrival difference distribution, which can be obtained from integrating the JTI in panels (a) and (b) over the time-sum variable. The result is plotted in figure 4(e), where the curve composed of multiple peaks corresponds to the spectrally unfiltered state and the other curve, which again may be thought of as an envelope, corresponds to the central peak of the JSI. The multiple peaks visible in the unfiltered time of arrival difference distribution are labeled as 0, ± 1, ± 2,... so that a value j > 0 means that the signal photon is emitted j cavity iterations after the idler photon, whereas a value j < 0 means that the signal photon is emitted j cavity iterations before the idler photon. The spacing between peaks corresponds to the cavity round trip time.

Zoom In Zoom Out Reset image size

Figures 4(c), (d) and (f) are similar to panels (a)–(b) and (e), except that they correspond to the Cs cavity source (where the two sources are otherwise identical to each other). Note that in this case the time of arrival difference distribution is asymmetric, where positive values do not occur, i.e. the signal photon in a given pair is always are emitted after the corresponding idler photon.

As was shown in section 2.2, we can obtain the JTI by numerical evaluation of equation (16). However, in this section we show that it is possible to obtain an expression in closed analytic form under certain approximations. These approximations include: (i) describing each cavity mode through a Gaussian function of the signal and idler frequencies, (ii) assuming that the SFWM mode widths are equal, i.e. δω_s = δω_i ≡ δω, and that the mode spacings are likewise equal, i.e. Δω_s = Δω_i ≡ Δω , and (iii) assuming that the JSA can be described only in terms of the pump bandwidth and the cavity mode structure, so that the sinc function in equation (2) can be replaced by unity. Also, let us assume that each of the signal and idler modes is filtered with rectangular spectral filters of width MΔω, centered around the particular cavity mode of interest. Thus, we can write the JSA in the form

$$\begin{eqnarray} \Upsilon ({\nu }_{\mathrm{s}},{\nu }_{\mathrm{i}})&=&{\mathrm{e}}^{-\frac{({\nu }_{\mathrm{s}}+{\nu }_{\mathrm{i}})^{2}}{2{\sigma }^{2}}}\sum _{l=-M}^{M}\sum _{m=-M}^{M}{\mathrm{e}}^{-\frac{({\nu }_{\mathrm{s}}-l \Delta \omega )^{2}}{\delta {\omega }^{2}}}\nonumber\\ &&\times ~{\mathrm{e}}^{-\frac{({\nu }_{\mathrm{i}}-m \Delta \omega )^{2}}{\delta {\omega }^{2}}}, \end{eqnarray} \tag{ 18 }$$

which has been expressed in terms of the frequency detunings ν_μ = ω_μ − ω_μ0 (with μ = s,i). The JTA, $\tilde {\Upsilon }({t}_{\mathrm{s}},{t}_{\mathrm{i}})$, can now be calculated analytically as the Fourier transform of ϒ(ν_s,ν_i). It is convenient to express the JTI in terms of the time-sum ${t}_{+}=({t}_{\mathrm{s}}+{t}_{\mathrm{i}})/\sqrt{2}$ and time-difference ${t}_{-}=({t}_{\mathrm{s}}-{t}_{\mathrm{i}})/\sqrt{2}$ variables, obtaining

$$\begin{eqnarray} {\tilde {S}}_{\mathrm{t}}({t}_{-},{t}_{+})&=&\exp \left (-\frac{{t}_{-}^{2}}{{\tau }_{\mathrm{c}}^{2}}-\frac{{t}_{+}^{2}}{{\tau }^{2}}\right )\nonumber\\ &&\times ~\frac{{\sin }^{2}\{ (M+\frac{1}{2})\frac{\Delta \omega }{\sqrt{2}}[{t}_{-}-\left (\frac{{\tau }_{\mathrm{c}}}{\tau }\right )^{2}{t}_{+}]\} }{{\sin }^{2}\{ \frac{\Delta \omega }{2\sqrt{2}}[{t}_{-}-\left (\frac{{\tau }_{\mathrm{c}}}{\tau }\right )^{2}{t}_{+}]\} }\nonumber\\ &&\times ~\frac{{\sin }^{2}\{ (M+\frac{1}{2})\frac{\Delta \omega }{\sqrt{2}}[{t}_{-}+\left (\frac{{\tau }_{\mathrm{c}}}{\tau }\right )^{2}{t}_{+}]\} }{{\sin }^{2}\{ \frac{\Delta \omega }{2\sqrt{2}}[{t}_{-}+\left (\frac{{\tau }_{\mathrm{c}}}{\tau }\right )^{2}{t}_{+}]\} }, \end{eqnarray} \tag{ 19 }$$

In equation (19), ${\tau }_{\mathrm{c}}=\sqrt{2}/\delta \omega$ is the temporal width along the t₋ direction, or correlation time, which defines the uncertainty in the time of emission difference. Note that this parameter is proportional to $\sqrt{\mathscr{F}}$ (see equation (13)), so that increasing the finesse leads to more signal and idler cavity round trips and hence to a greater correlation time. Parameter τ in equation (19), given by $\tau ={\tau }_{\mathrm{c}}\sqrt{\delta {\omega }^{2}+{\sigma }^{2}}/\sigma$, represents the temporal width along the t₊ direction.

The expression which we have derived for the SFWM flux (see equation (21)) is in terms of a two-dimensional integral. While flux values may be obtained through numerical integration of equation (21), as was done for figure 6, we may gain additional physical insight from a crude approximation of the flux as N = AH, where A is the area in {ω_s,ω_i} space in which the joint spectrum has an appreciable magnitude, and H is the maximum value of the joint intensity, within this area. For our present analysis we are interested in the SFWM flux obtained in a χ⁽³⁾ cavity, N, normalized by the corresponding flux in an equivalent source without a cavity, likewise expressed as N_nc = A_ncH_nc. The quantity ξ may then be approximated as ξ ≈ ad, where a ≡ H/H_nc and d ≡ A/A_nc.

Because the flux enhancement effect occurs for a single cavity mode in a manner qualitatively similar to that for multiple cavity modes (see figure 6), it is sufficient to analyze the former case in order to obtain an understanding of the relevant physics. Thus, in what follows we study the behavior of ξ as a function of the pump bandwidth, assuming that the signal and idler photons are filtered using rectangular spectral filters of widths Δω_s and Δω_i, respectively, so that a single cavity mode is retained. Note that while for non-degenerate SFWM the intra-mode spacing Δω and the mode width δω differ between the two SFWM modes (for the specific design in section 3 this variation is near 5%), for the present analysis we will assume that Δω_s = Δω_i and δω_s = δω_i. We are interested in the three zones (i) ${\sigma }_{\mathrm{I}}\gt \sqrt{2}\Delta \omega$, (ii) $\delta \omega \lt {\sigma }_{\mathrm{I}}\lt \sqrt{2}\Delta \omega$, and (iii) σ_I < δω considered before.

Let us first analyze in this manner a Csi cavity. In this case, it can be shown that parameter a is given by

$$\begin{eqnarray} &&a=\left (\frac{1+\vert {r}_{2}\vert }{1-\vert {r}_{2}\vert }\right )^{2}. \end{eqnarray} \tag{ 24 }$$

In figure 7(a) we show schematically the cavity mode structure. Each circle corresponds to a cavity mode of width δω, separated from each neighboring mode, along the signal and/or idler frequency axes, by Δω. The spectral filters of width Δω correspond to retaining only the flux produced within one of the squares in the grid shown. The area shaded in red corresponds to the flux emitted with a cavity source, while the (square) area shaded in blue corresponds to the flux emitted with an equivalent source without a cavity.

Zoom In Zoom Out Reset image size

In region 1, A is the area of a disk of diameter δω, i.e. A = πδω²/4, while A_nc is the area of a square of dimension Δω, i.e. A_nc = Δω², so that

$$\begin{eqnarray} &&{\xi }_{1}=a\frac{\pi \delta {\omega }^{2}}{4 \Delta {\omega }^{2}}=\frac{(1+\vert {r}_{2}\vert )^{2}}{4 \pi \vert {r}_{2}\vert }, \end{eqnarray} \tag{ 25 }$$

which is independent of σ_I. In region 2, A = πδω²/4, while ${A}_{\mathrm{nc}}={\sigma }_{\mathrm{I}}(\sqrt{2}\Delta \omega -{\sigma }_{\mathrm{I}}/2)$, so that

$$\begin{eqnarray} &&{\xi }_{2}=\frac{a \pi \delta {\omega }^{2}}{4{\sigma }_{\mathrm{I}}(\sqrt{2}\Delta \omega -{\sigma }_{\mathrm{I}}/2)}. \end{eqnarray} \tag{ 26 }$$

In region 3, we approximate A as A ≈ δωσ_I, an approximation which becomes better for smaller σ_I, while ${A}_{\mathrm{nc}}={\sigma }_{\mathrm{I}}(\sqrt{2}\Delta \omega -{\sigma }_{\mathrm{I}}/2)$, so that

$$\begin{eqnarray} &&{\xi }_{3}=\frac{a \delta \omega }{\sqrt{2}\Delta \omega -{\sigma }_{\mathrm{I}}/2}. \end{eqnarray} \tag{ 27 }$$

Note that the last two expressions for ξ (equations (26) and (27)) depend on σ_I and yield the flux enhancement described below. This leads to an expression for the flux enhancement, E, defined as the quotient ξ₁/ξ₃, with σ_I → 0. We thus obtain

$$\begin{eqnarray} &&E=\sqrt{2\mathscr{F}}, \end{eqnarray} \tag{ 28 }$$

making it clear that the flux enhancement increases with the coefficient of finesse.

Let us now turn our attention to a cavity source which is resonant only for one of the SFMW modes, i.e. a Cs or Ci cavity. In this case the parameter a is given by

$$\begin{eqnarray} &&a=\frac{1+\vert {r}_{2}\vert }{1-\vert {r}_{2}\vert }. \end{eqnarray} \tag{ 29 }$$

As can be seen in figure 7(b), which is similar to figure 7(a) drawn for the Cs cavity, in all three of the regions considered above, A is the area of a parallelogram with height δω and base $\sqrt{2}\sigma$, while A_nc is also the area of a parallelogram with height Δω and base $\sqrt{2}\sigma$. Thus, the ratio between the flux emitted from the cavity and the flux emitted from SFWM without a cavity is the same in the three regions and is given by

$$\begin{eqnarray} &&\xi =\frac{a \delta \omega }{\Delta \omega }. \end{eqnarray} \tag{ 30 }$$

This result indicates that in a Cs cavity there is no an enhancement in the emitted flux due to the cavity, as was the case for a Csi cavity. This behavior can be appreciated in figure 6.

In our discussion of the SFWM flux, we have so far focused on the flux attainable in a SFWM cavity source relative to an equivalent source without a cavity. It is likewise important to study the flux in absolute terms, which our analysis in section 4 and in particular equation (21) permits.

While much of the behavior of the relative flux discussed so far is qualitatively identical to the behavior expected for a SPDC χ⁽²⁾ cavity source, the absolute flux leads to some considerably different behaviors between the χ⁽²⁾ and χ⁽³⁾ cases. This difference is related to the fact that, for SFWM, two pump photons are annihilated in each generation event rather than just one for SPDC. This implies that, within the phasematching bandwidth σ_PM (i.e. for σ_I ≲ σ_PM), the SFWM flux scales linearly with the pump bandwidth σ_I while the SPDC flux remains constant with respect to σ_I for σ_I ≲ σ_PM. In physical terms, each pump frequency of one pump photon may participate in the SFWM process together with all pump frequencies of the other pump photon, limited by phasematching, so that increasing the pump bandwidth leads to a greater range of phasematched frequency combinations, and hence to a greater flux. In contrast, in the case of SPDC, each pump frequency may be considered to act independently from other frequencies leading to a constant flux versus σ dependence, for a constant average pump power.

In figure 8(a) we present the absolute SFWM flux, assuming a Csi cavity, plotted as a function of the pump bandwidth σ_I, for a number of different values of the reflectivity |r₂|, in particular for |r₂| = 0.8, |r₂| = 0.9 and |r₂| = 0.99, with other source parameters selected to be identical to those assumed for figure 6. In this figure we have also indicated the cavity mode width δω value for each of these reflectivities and the $\sqrt{2}\Delta \omega$ value. Note that the effect of the cavity is that the flux versus σ_I dependence is linear for σ_I ≲ δω, climbs more slowly for ${\sigma }_{\mathrm{I}}\gtrsim \sqrt{2}\delta \omega$, and thereafter reaches a plateau at a value of σ_I, which for large is close to δω. Thus, while the maximum cavity-induced flux enhancement occurs for σ_I → 0, as is clear from figure 6, the maximum absolute flux occurs within the above mentioned plateau. For a large , the flux can be maximized if the pump bandwidth satisfies σ_I ≳ δω. Note that in the case of Cs cavity, the absolute flux versus σ_I curves (not shown) yield essentially the same curve as the case without cavity shown in figure 8(a).

Zoom In Zoom Out Reset image size

As a particular example, let us calculate the absolute flux for the specific source design of section 3. This source involves degenerate pumps at λ_p = 1.064 μm, with SFWM wavelengths λ_s = 0.852 μm and λ_i = 1.417 μm. We assume that the cavity source is based on a photonic crystal fiber of length L = 1 cm, with a core radius of r = 0.68 μm and air-filling fraction of f = 0.5. We assume a reflectivity of |r₂| = 0.998, which corresponds to a coefficient of finesse of  = 9.98 × 10⁵, and a cavity mode width of δω = 2π × 5.22 MHz, which defines resulting SFWM emission bandwidth and is matched to the D2 line transition of cesium. We assume an average pump power of 300 mW, with a pump bandwidth σ_I selected to have a value σ_I = 5δω = 0.164 GHz which fulfils the condition σ_I ≳ δω of the previous paragraph. This value of σ_I corresponds to a pulse duration of 16.9 ns, and we assume a repetition rate of 0.1 MHz. The resulting photon pair flux can then determined by numerical integration of equation (21): 1.03 × 10⁹ photon pairs s⁻¹.