# Precise Photon Correlation Measurement of a Chaotic Laser
## 2 High Order Correction of \(g^{(2)}(\tau)\)  
Theoretically, second order photon correlation of \(g^{(2)}(\tau)\) can be obtained from an ideal photon pair time interval distribution \(P_{1}(\tau)\). Using the self-convolution method, one can obtain any desired high order \(n\), and the higher \(n\) of \(g^{(2)}_{n}(\tau)\) is, the more accurate \(g^{(2)}_{n}(\tau)\) tends to the ideal \(g^{(2)}(\tau)\). But due to the actual operation capacity of data processing and the difficulty of convolving complex form to very high order, we reasonably convolve \(g^{(2)}_{n}(\tau)\) to the ninth order so the relative error is small enough to obtain high accuracy.  
In our experiment, photon pair time interval distribution is collected by single photon counters and the time distribution is \(D_{1}(\tau)\). Furthermore, \(g^{(2)}(\tau)\) can be calculated from the self-convolution of \(D_{1}(\tau)\).  
The second order photon correlation \(g^{(2)}(\tau)\) has a proportional relation to \(G(\tau)\), as follows:  
\[G(\tau)=\widetilde{I}g^{(2)}(\tau), \tag{1}\]  
where \(\widetilde{I}\) is the average photon counting rate per time bin of the light field. \(G(\tau)\) is the histogram of photons at delay time \(\tau\) between two photon detection events. The relation between \(G(\tau)\) and \(P_{1}(\tau)\) is given by:  
\[G(\tau)=P_{1}(\tau)+P_{1}(\tau)*P_{1}(\tau)+\ldots\ldots=\sum_{n=1}^{\infty}P_ {n}(\tau), \tag{2}\]  
where \(P_{1}(\tau)\) is an ideal photon pair time interval distribution of light field which can be obtained based on HBT experiment, and \(P_{n}(\tau)\) is \(n\)th order self-convolution of \(P_{1}(\tau)\). When \(P_{1}(\tau)\) is less than one, the sum of \(P_{n}(\tau)\) is convergence [40], then we can obtain:  
\[P_{1}(\tau)=L^{-1}(\frac{L(G(\tau))}{1+L(G(\tau))}), \tag{3}\]  
where \(L\) denotes the Laplace transformation, and \(L^{-1}\) denotes the inverse Laplace transformation.  
When the above theory is applied to a Lorentzian chaotic laser field, we can get the relation between \(G(\tau)\) and \(P_{n}(\tau)\) of a chaotic laser. The first order correlation of a Lorentzian chaotic laser is as follows:  
\[g^{(1)}(\tau)=e^{-\frac{|t|}{t_{c}}}. \tag{4}\]  
The relation between \(g^{(2)}(\tau)\) and \(g^{(1)}(\tau)\) for a Lorentzian chaotic laser is:  
\[g^{(2)}(\tau)=1+\big{|}g^{(1)}(\tau)\big{|}^{2}. \tag{5}\]  
Using Equations (4) and (5), we obtain:  
\[g^{(2)}(\tau)=1+\big{|}e^{-|\tau|/\tau_{c}}\big{|}^{2}. \tag{6}\]  
According to Equations (1) and (6), we obtain:  
\[G(\tau)=\widetilde{I}(1+\big{|}e^{-|\tau|/\tau_{c}}\big{|}^{2}). \tag{7}\]  
The relation between \(g^{(2)}(\tau)\) and \(P_{n}(\tau)\) is shown as follows.  
\(P_{2}(\tau)\) is the self-convolution of \(P_{1}(\tau)\):  
\[P_{2}(\tau)=\int_{0}^{+\infty}P_{1}(\tau)P_{1}(t-\tau)dt=P_{1}(\tau)*P_{1}( \tau). \tag{8}\]  
\(P_{3}(\tau)\) is the convolution of \(P_{1}(\tau)\) and \(P_{2}(\tau)\):  
\[P_{3}(\tau)=\int_{0}^{+\infty}P_{2}(\tau)P_{1}(t-\tau)dt=P_{2}(\tau)*P_{1}( \tau). \tag{9}\]  
\(P_{n}(\tau)\) is the convolution of \(P_{1}(\tau)\) and \(P_{n-1}(\tau)\):  
\[P_{n}(\tau)=\int_{0}^{+\infty}P_{n-1}(\tau)P_{1}(t-\tau)dt=P_{n-1}(\tau)*P_{1} (\tau). \tag{10}\]  
The above analysis basically solves the high order correction \(g_{n}^{(2)}(\tau)\) of the chaotic laser in theory and experiment. One can also use this method to analyze the error caused by the variations of the mean photon intensity and the coherence time of the laser. In addition, high order correction of \(g^{(2)}(\tau)\) for coherent light can be achieved and the \(g^{(2)}(\tau)\) is perfectly equal to one.  
Using Equation (14), \(g_{n}^{(2)}(\tau)\) is calculated to ninth order, and higher order terms than ninth can be omitted. Relative error \(\delta\) varying with the delay time \(\tau\) at the correction order of nine is calculated as:  
\[\delta=\frac{\left|g_{n}^{(2)}(\tau)-g^{(2)}(\tau)\right|}{g^{(2)}(\tau)}\times 1 00\%. \tag{18}\]