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Conference on Lasers and Electro-Optics

Absolute rates of Spontaneous Parametric Down Conversion into a single transverse Gaussian mode

Ling, Lamas-Linares, Kurtsiefer

Published Jan 1, 2008• 0 citations
Publisher

Abstract

10.1109/QELS.2008.4552947
DOI
10.1109/qels.2008.4552947

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Comments on: "Introduction to the Absolute Brightness and Number Statistics in Spontaneous Parametric Down-Conversion" (2019 J. Opt. 21, 043501)

James Schneeloch

These comments contain two primary improvements to the aforementioned article: Where previously mentioned in passing in section 3.4 of our article, we now provide, using the same basic steps as in Bennink's paper, [Phys. Rev. A 81, 053805 (2010)] (with some expanded discussion), a full derivation for the absolute generation rate of photon pairs from Spontaneous-Parametric Down-Conversion (SPDC) when using a focused gaussian pump beam as input, and collecting the output biphotons in a pair of focused gaussian collection modes. We do this to both show consistency between our formalism and his treatment, and to be able to use it to predict the absolute brightness of photon pair generation in these sources. In doing so, our formula differs from that in [Phys. Rev. A 81, 053805 (2010)] by only a constant factor based on the ratio of products of indices of refraction. Second, we correct an abuse of notation in our main work in which the quantization dimensions of the cavity modes of the electromagnetic field are conflated with the dimensions of the nonlinear medium. While this does not meaningfully affect the final formulas we obtain for the generation rates in the paper (adding a dimensionless correction factor of order unity), we use the gaussian-mode SPDC derivation here to show how the factors of the quantization dimensions disappear when converting from a discrete-momentum basis of rectangular cavity modes to a continuous frequency basis of gaussian spatial modes. In doing this, we find that applying this correction factor to our previous theoretical predictions for the photon pair generation rates brings them closer to the obtained experimental values.

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Introduction to the absolute brightness and number statistics in spontaneous parametric down-conversion

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Abstract As a tutorial, we examine the absolute brightness and number statistics of photon pairs generated in spontaneous parametric down-conversion (SPDC) from first principles. In doing so, we demonstrate how the diverse implementations of SPDC can be understood through a single common framework, and use this to derive straightforward formulas for the biphoton generation rate (pairs per second) in a variety of different circumstances. In particular, we consider the common cases of both collimated and focused Gaussian pump beams in a bulk nonlinear crystal, as well as in nonlinear waveguides and micro-ring resonators. Furthermore, we examine the number statistics of down-converted light using a non-perturbative approximation (the multi-mode squeezed vacuum), to provide quantitative formulas for the relative likelihood of multi-pair production events, and explore how the quantum state of the pump affects the subsequent statistics of the down-converted light. Following this, we consider the limits of the undepleted pump approximation, and conclude by performing experiments to test the effectiveness of our theoretical predictions for the biphoton generation rate in a variety of different sources.

arXiv:1807.10885•Journal of Optics•Feb 8, 2019•88 citations
We provide an estimate of the absolute values of the emission rate of photon pairs produced by spontaneous parametric down-conversion in a bulk crystal when all interacting fields are in single transverse Gaussian modes. Both collinear and noncollinear configurations are covered, and we arrive at a fully analytical expression for the collinear case. Our results agree reasonably well with values found in typical experiments, which allows this model to be used for understanding the dependency on the relevant experimental parameters.
These comments contain two primary improvements to the aforementioned article: Where previously mentioned in passing in section 3.4 of our article, we now provide, using the same basic steps as in Bennink's paper, [Phys. Rev. A 81, 053805 (2010)] (with some expanded discussion), a full derivation for the absolute generation rate of photon pairs from Spontaneous-Parametric Down-Conversion (SPDC) when using a focused gaussian pump beam as input, and collecting the output biphotons in a pair of focused gaussian collection modes. We do this to both show consistency between our formalism and his treatment, and to be able to use it to predict the absolute brightness of photon pair generation in these sources. In doing so, our formula differs from that in [Phys. Rev. A 81, 053805 (2010)] by only a constant factor based on the ratio of products of indices of refraction. Second, we correct an abuse of notation in our main work in which the quantization dimensions of the cavity modes of the electromagnetic field are conflated with the dimensions of the nonlinear medium. While this does not meaningfully affect the final formulas we obtain for the generation rates in the paper (adding a dimensionless correction factor of order unity), we use the gaussian-mode SPDC derivation here to show how the factors of the quantization dimensions disappear when converting from a discrete-momentum basis of rectangular cavity modes to a continuous frequency basis of gaussian spatial modes. In doing this, we find that applying this correction factor to our previous theoretical predictions for the photon pair generation rates brings them closer to the obtained experimental values.
Abstract As a tutorial, we examine the absolute brightness and number statistics of photon pairs generated in spontaneous parametric down-conversion (SPDC) from first principles. In doing so, we demonstrate how the diverse implementations of SPDC can be understood through a single common framework, and use this to derive straightforward formulas for the biphoton generation rate (pairs per second) in a variety of different circumstances. In particular, we consider the common cases of both collimated and focused Gaussian pump beams in a bulk nonlinear crystal, as well as in nonlinear waveguides and micro-ring resonators. Furthermore, we examine the number statistics of down-converted light using a non-perturbative approximation (the multi-mode squeezed vacuum), to provide quantitative formulas for the relative likelihood of multi-pair production events, and explore how the quantum state of the pump affects the subsequent statistics of the down-converted light. Following this, we consider the limits of the undepleted pump approximation, and conclude by performing experiments to test the effectiveness of our theoretical predictions for the biphoton generation rate in a variety of different sources.