Optimal coupling of entangled photons into single-mode optical fibers

R. Andrews; E. R. Pike; Sarben Sarkar

doi:10.1364/OPEX.12.003264

Optics Express
Vol. 12,
Issue 14,
pp. 3264-3269
(2004)
•https://doi.org/10.1364/OPEX.12.003264

Optimal coupling of entangled photons into single-mode optical fibers

R. Andrews, E. R. Pike, and Sarben Sarkar

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R. Andrews, E. R. Pike, and Sarben Sarkar, "Optimal coupling of entangled photons into single-mode optical fibers," Opt. Express 12, 3264-3269 (2004)

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Abstract

We present a consistent multimode theory that describes the coupling of single photons generated by collinear Type-I parametric down-conversion into single-mode optical fibers. We have calculated an analytic expression for the fiber diameter which maximizes the pair photon count rate. For a given focal length and wavelength, a lower limit of the fiber diameter for satisfactory coupling is obtained.

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Fig. 1.

Fig. 1. Schematic of the experimental setup to determine pair photon coupling in fibers

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A^{(2)} = 〈 {\vec{E}}_{H}^{+} ({\vec{r}}_{1}, t_{1}) {\vec{E}}_{H}^{+} ({\vec{r}}_{2}, t_{2}) 〉

\int d^{3} r_{3} \int d^{3} k_{1} \int d^{3} k_{2} U_{{\vec{k}}_{1} {\vec{λ}}_{1}}^{*} ({\vec{r}}_{3}) U_{{\vec{k}}_{2} λ_{2}}^{*} ({\vec{r}}_{3}) U_{{\vec{k}}_{0} λ_{0}} ({\vec{r}}_{3}) f_{p} ({\vec{r}}_{3}) {(\frac{ℏ ω_{k_{0}}}{2 ε_{0}})}^{\frac{1}{2}} {(\frac{ℏ ω_{k_{1}}}{2 ε_{0}})}^{\frac{1}{2}} {(\frac{ℏ ω_{k_{2}}}{2 ε_{0}})}^{\frac{1}{2}}

\times 〈 α_{k_{0}}, 0 ∣ {\vec{E}}_{I}^{(+)} ({\vec{r}}_{2}, t_{2}) {\vec{E}}_{I}^{(+)} ({\vec{r}}_{1}, t_{1}) ∣ ω_{k_{0}}, k_{1}, k_{2} 〉 δ (ω_{k_{1}} - ω_{k_{2}} - ω_{k_{0}})

{\vec{E}}^{+} (\vec{r}, t) = i \sum_{λ λ'} \int d^{3} k' \int d^{3} k {(\frac{ℏ ω_{k}}{2 ε_{0}})}^{\frac{1}{2}} {\vec{ε}}_{\vec{k} λ} β_{\vec{k'} λ', \vec{k} λ} U_{\vec{k} λ}^{f} (\vec{r}) e^{- i ω t} a_{\vec{k} λ}

β_{\vec{k}' λ' \vec{k} λ} = \int d x d y U_{\vec{k}' λ'}^{in} (x, y) U_{\vec{k} λ}^{f *} (x, y)

\int d^{3} k_{1} \int d^{3} k_{2} {\tilde{f}}_{p} ({\vec{k}}_{1 t} + {\vec{k}}_{2 t}) \sin c (Δ_{k_{1 z} k_{2 z}} \frac{d}{2}) β_{k_{1} λ_{1}} β_{k_{2} λ_{2}} U^{f} ({\vec{r}}_{1}) U^{f} ({\vec{r}}_{2})

\times {(\frac{ℏ ω_{k_{1}}}{2 ε_{0}})}^{\frac{1}{2}} {(\frac{ℏ ω_{k_{2}}}{2 ε_{0}})}^{\frac{1}{2}} e^{- i ω_{k_{1}} t_{1}} e^{- i ω_{k_{2}} t_{2}} δ (ω_{k_{0}} - ω_{k_{1}} - ω_{k_{2}})

f ({\vec{r}}_{t}) \propto \exp (- \frac{x^{2} + y^{2}}{w_{p}})

{\tilde{f}}_{p} ({\vec{k}}_{1 t} + {\vec{k}}_{2 t}) \propto \exp (- \frac{1}{4} w_{p}^{2} v^{2} {(ω_{k_{1}} - ω_{k_{2}})}^{2} \sin^{2} θ^{*})

U_{k λ}^{in} (x, y) = \frac{1}{i k d'} e^{i k (d' + d)} e^{i \frac{k}{2 d'} (x^{2} + y^{2})} \int_{0}^{2 π} \int_{0}^{R} r d θ d r (e^{i (\frac{k}{2 d'} - \frac{k}{2 f}) r^{2}} e^{- i k (\frac{x \cos θ}{d'} + \frac{y \sin θ}{d'}) r})

U^{f} (x, y) = \frac{1}{w_{0} \sqrt{π}} \exp (- \frac{x^{2} + y^{2}}{2 w_{0}^{2}})

β_{k λ} \propto \frac{w_{0}^{2} {d'}^{2} + i k w_{0}^{4} d'}{k d' w_{0} ({d'}^{2} + k^{2} w_{0}^{4})} \frac{(1 - e^{(- A_{k} + i B_{k}) R^{2}})}{(- A_{k} + i B_{k})}

A_{k} = \frac{k^{2} w_{0}^{2}}{2 ({d'}^{2} + k^{2} w_{0}^{4})}

B_{k} = (\frac{1}{2 d'} - \frac{1}{2 f}) k - \frac{k^{3} w_{0}^{4}}{2 d' ({d'}^{2} + k^{2} w_{0}^{4})}

A^{(2)} \propto \int_{- \infty}^{\infty} e^{- i τ x \frac{x^{2}}{Δ^{2}}} {(1 - e^{(a + b x)})}^{2} d x

a = (- \frac{k^{* 2} w_{0}^{2}}{2 f^{2}} - i \frac{k^{* 3} w_{0}^{4}}{2 f^{3}}) R^{2}

b = (- \frac{k^{*} w_{0}^{2} n_{g}}{c f^{2}} - i \frac{3}{2} \frac{n_{g} k^{* 2} w_{0}^{4}}{2 c f^{3}}) R^{2}

C \propto (1 - 4 e^{- y} + 6 e^{- 2 y} - 4 e^{- 3 y} + e^{- 4 y})

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Equations (18)

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