High-order temporal coherences of&#x2028;chaotic and laser light

Martin J. Stevens; Burm Baek; Eric A. Dauler; Andrew J. Kerman; Richard J. Molnar; Scott A. Hamilton; Karl K. Berggren; Richard P. Mirin; Sae Woo Nam

doi:10.1364/OE.18.001430

Optics Express
Vol. 18,
Issue 2,
pp. 1430-1437
(2010)
•https://doi.org/10.1364/OE.18.001430

High-order temporal coherences of chaotic and laser light

Martin J. Stevens, Burm Baek, Eric A. Dauler, Andrew J. Kerman, Richard J. Molnar, Scott A. Hamilton, Karl K. Berggren, Richard P. Mirin, and Sae Woo Nam

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Martin J. Stevens, Burm Baek, Eric A. Dauler, Andrew J. Kerman, Richard J. Molnar, Scott A. Hamilton, Karl K. Berggren, Richard P. Mirin, and Sae Woo Nam, "High-order temporal coherences of chaotic and laser light," Opt. Express 18, 1430-1437 (2010)

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Table of Contents Category
- Coherence and Statistical Optics
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About this Article
History
- Original Manuscript: November 18, 2009
- Revised Manuscript: December 23, 2009
- Manuscript Accepted: December 26, 2009
- Published: January 12, 2010

Abstract

We demonstrate a new approach to measuring high-order temporal coherences that uses a four-element superconducting nanowire single-photon detector. The four independent, interleaved single-photon-sensitive elements parse a single spatial mode of an optical beam over dimensions smaller than the minimum diffraction-limited spot size. Integrating this device with four-channel time-tagging electronics to generate multi-start, multi-stop histograms enables measurement of temporal coherences up to fourth order for a continuous range of all associated time delays. We observe high-order photon bunching from a chaotic, pseudo-thermal light source, measuring maximum third- and fourth-order coherence values of 5.87 ± 0.17 and 23.1 ± 1.8, respectively, in agreement with the theoretically predicted values of 3! = 6 and 4! = 24. Laser light, by contrast, is confirmed to have coherence values of approximately 1 for second, third and fourth orders at all time delays.

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Fig. 1 Scanning-electron microscope image of the four-element SNSPD, with nanowire elements 0-3 traced out in color. Each element consists of a ~5 nm-thick × 80 nm-wide NbN nanowire on a sapphire substrate, with 60 nm gaps between wires. The 9.4 µm-diameter active area is well matched to the spatial mode of a single mode optical fiber, the cleaved end of which is held within ~10 µm of the detector surface. The interleaved design ensures that all four elements equally sample this spatial mode.

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Fig. 2 Measured n^th -order temporal coherences for n = 2 (○), 3 (▲) and 4 (■). (a) Chaotic source data. The magnitude of photon bunching scales roughly as n!, as can be seen from the expected peak g ⁽²⁾, g ⁽³⁾ and g ⁽⁴⁾ values of 2 (dashed green line), 6 (dashed red line) and 24 (top axis). (b) Laser source data, also showing the expected value of 1 (black line) and the mean measured g ⁽⁴⁾ value of 1.011 (dashed blue line). In both (a) and (b), the g ⁽²⁾ data are displayed as a function of τ, while the g ⁽³⁾ and g ⁽⁴⁾ data are plotted against parameterized delays that measure temporal distance from the origin, τ_P ₃ = ± (τ ₁ ² + τ ₂ ²)^1/2 and τ_P4 = ± (τ ₁ ² + τ ₂ ² + τ ₃ ²)^1/2, respectively, where ± is determined by the sign of τ ₁. These trace out cross sections from (τ ₁, τ ₂) = (−6/√2 μs, 6/√2 μs) to (6/√2 μs, −6/√2 μs) and (τ ₁, τ ₂, τ ₃) from (−4 μs, 2 μs, 4 μs) to (4 μs, −2 μs, −4 μs).

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Fig. 3 (a) Measured third-order coherence from the chaotic source, where both color and height indicate measured value of g ⁽³⁾. The cross-section in Fig. 2(a) samples these data along a diagonal line (not shown) extending from the far left corner to the far right corner as plotted here. (b) Calculated third-order coherence for a chaotic source derived from an ideal Gaussian scattering process with a coherence time of 900 ns, as discussed in the text.

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Fig. 4 Four frames from a movie (Media 1) of fourth-order coherence data (left) and theory (right) for the chaotic source for four values of τ ₃. Color bar at the bottom shows g ⁽⁴⁾ values.

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Equations (5)

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g^{(2)} (τ) = \frac{〈 {\hat{a}}^{†} (t) {\hat{a}}^{†} (t + τ) \hat{a} (t + τ) \hat{a} (t) 〉}{{〈 {\hat{a}}^{†} (t) \hat{a} (t) 〉}^{2}},

g^{(3)} (τ_{1}, τ_{2}) = \frac{〈 {\hat{a}}^{†} (t) {\hat{a}}^{†} (t + τ_{1}) {\hat{a}}^{†} (t + τ_{2}) \hat{a} (t + τ_{2}) \hat{a} (t + τ_{1}) \hat{a} (t) 〉}{{〈 {\hat{a}}^{†} (t) \hat{a} (t) 〉}^{3}},

g^{(4)} (τ_{1}, τ_{2}, τ_{3}) = \frac{〈 {\hat{a}}^{†} (t) {\hat{a}}^{†} (t + τ_{1}) {\hat{a}}^{†} (t + τ_{2}) {\hat{a}}^{†} (t + τ_{3}) \hat{a} (t + τ_{3}) \hat{a} (t + τ_{2}) \hat{a} (t + τ_{1}) \hat{a} (t) 〉}{{〈 {\hat{a}}^{†} (t) \hat{a} (t) 〉}^{4}},

g^{(3)} (τ_{1}, τ_{2}) = \frac{〈 I (t) I (t + τ_{1}) I (t + τ_{2}) 〉}{{〈 I (t) 〉}^{3}}

g^{(4)} (τ_{1}, τ_{2}, τ_{3}) = \frac{〈 I (t) I (t + τ_{1}) I (t + τ_{2}) I (t + τ_{3}) 〉}{{〈 I (t) 〉}^{4}}

Abstract

Supplementary Material (1)

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Equations (5)

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