# Precise Photon Correlation Measurement of a Chaotic Laser
## 5 Influences of Detector Time Resolution and High Order Omitted Terms  
In our experiment, the resolution time of the detection (65 ps) was not significantly small compared to the coherence time (\(\sim\)0.5 ns) of the chaotic laser, resulting in a little fluctuation of measured \(g^{(2)}(\tau)\). In Figure 5a, the experimental results of \(g^{(2)}_{n}(\tau)\) within 100 ns delay time is shown and the magenta curve represents the original photon pair time interval distribution. The original experimental data is the same as those used in Figure 4b. The bottom-up colored curves indicate the increasing order corrections of second order photon correlation. The orange curve is the third order correction of \(g^{(2)}(\tau)\), and the others are fifth, seventh, and ninth order corrections of \(g^{(2)}(\tau)\). For an accurate measurement of photon correlation, a very low photon flux rate \(\overline{l}\) was required to ensure \(\overline{l}\tau_{c}<1\)[39]. The counting rate of the SPD was controlled below 0.3 Mcounts/s by using the VA2 and the overall detection efficiency was 25%. In Figure 5a, the counting rate was 270 kcounts/s and the dead time of the SPD was 4 \(\upmu\)s. Within 100 ns sampling time (i.e., \(\tau\) = 100 ns), the incident light intensity was estimated to be about \(4\times 10^{7}\) photons/s. Figure 5b shows the theoretical results when \(\tau_{c}\) was 0.5 ns and the incident light intensity was \(4\times 10^{7}\) photons/s. The experimental results are in good agreement with the theory. Relative Error of \(g^{(2)}_{n}(\tau)\) With Mean Photon Intensity and Coherence Time of the Chaotic Laser  
The coherence time of our experiment was below 1 ns and we can set the maximum coherence time in the theoretical analysis. Following this, \(g^{(2)}(\tau)\) was obtained by using Equation (6), which is independent on the mean photon intensity. Furthermore, according to Equation (18), it is found that the mean photon intensity and coherence time have effects on the relative error. The maximum photon intensity in our experiment did not exceed 0.05 photons/ns. Given this finding, we changed the mean photon intensity from 0.03 photons/ns to 0.05 photons/ns. For the low order correction of \(g^{(2)}(\tau)\), it cannot provide sufficient information and accuracy according to Equation (2). For the ninth order correction, there was almost no difference between \(g^{(2)}_{9}(\tau)\) and \(g^{(2)}(\tau)\) and the loss information can be ignored. Figure 6 shows the relative error of \(g^{(2)}_{9}(\tau)\) for photon intensity changes from 0.03 photons/ns to 0.05 photons/ns and different delay times with the ninth order correction. The relative error varied with the photon intensity and delay time. When the delay time is shorter than 40 ns the relative error can be ignored, while the relative error is increased when the delay time is close to 100 ns. It should be noted that higher order correction can reduce the relative error for longer delay time. In Figure 6, it is also indicated that larger photon intensity brings bigger error. But when the photon intensity is too low, the photon pair time interval distribution contains a lot of dark noise that deteriorates the detection performance. Figure 5: The (**a**) experimental and (**b**) theoretical results of \(g^{(2)}_{n}(\tau)\) within the delay time 100 ns. Following this, we theoretically analyzed the coherence time from 0.3 ns to 0.7 ns under the condition that the photon intensity was near \(4\times 10^{7}\) photons/s. Figure 7 shows the relative error as functions of the coherence time \(\tau_{c}\) and the delay time \(\tau\). The coherence time \(\tau_{c}\) varied from 0.3 ns to 0.7 ns and the delay time \(\tau\) was within 100 ns. In this case, corresponding to our experimental condition, the relative error did not exceed \(5\%_{\infty}\) within 50 ns delay time. It is worth noting that a long \(\tau_{c}\) leads to big relative error, but the change of relative error was subtle.