Precise Photon Correlation Measurement of a Chaotic Laser

The second order photon correlation g^(2)(tau) of a chaotic optical-feedback semiconductor laser is precisely measured using a Hanbury Brown-Twiss interferometer. The accurate g^(2)(tau) with non-zero delay time is obtained experimentally from the photon pair time interval distribution through a ninth-order self-convolution correction. The experimental results agree well with the theoretical analysis. The relative error of g^(2)(tau) is no more than 0.005 within 50 ns delay time. The bunching effect and coherence time of the chaotic laser are measured via the precise photon correlation technique. This technique provides a new tool to improve the accuracy of g^(2)(tau) measurement and boost applications of quantum statistics and correlation.


Introduction
Semiconductor lasers subject to external optical feedback exhibit a rich variety of nonlinear dynamical behaviors and are used to generate high-dimensional chaotic laser [1,2]. This configuration has attracted great interest for a wide range of applications, like optical chaos communication [3][4][5][6][7], secure key distribution [8], high-speed physical random number generation [9][10][11][12], chaos-based optical computing [13] and sensing [14][15][16]. It is fundamentally important to understand the underlying physical mechanisms of chaotic laser, and practically useful to improve the laser performance and motivate its applications. Previous researches mainly focused on clarifying intensity statistics and autocorrelation (AC) of chaotic lasers to characterize the chaotic processes [17][18][19][20][21].Intensity statistics are closely relevant to the extractible rate of random number [9,10,22] and the AC is a good indicator of chaotic modulating bandwidth in optical chaos communications [4,19]. However, the macro-scale intensity statistics and AC are not sufficient to reveal all properties of a given chaotic laser, and there is also a significant discrepancy between experimental and theoretical probability density distributions of the laser intensity [19]. Recent researches reveal that quantum correlation is more accurate in assessing statistical properties and more sensitive to control parameters compared to AC function [23,24]. However, the previous research is concentrated on the properties of the quantum dot laser in the low-intensity (low-gain) situation, and the bunching effect of the chaotic laser, i.e., g (2) (0)> 1 at τ = 0, is only revealed in the fully developed chaotic (high-gain) regime. Studies on high order photon correlation of high-dimensional chaotic lasers are sparse, especially second order photon correlation g (2) (τ) at non-zero delay time.
The landmark experiment on photon correlation was first conducted by Hanbury Brown and Twiss (HBT), demonstrating spatial second order photon correlation g (2) of a thermal light [25]. Soon afterwards, this experiment inspired Glauber's seminal work on quantum optics theory, which described the photon correlation of different light fields by correlation functions within quantum statistics [26][27][28]. The photon correlation g (2) is fundamentally different from the first order correlation and is harnessed in many applications, such as photon bunching and anti-bunching measurement [29][30][31][32], spatial interference [33,34], ghost imaging [35][36][37], the azimuthal HBT effect [38], single photon detection [39], etc. The g (2) (τ) also carries a wealth of information on the statistical probability of different photons arriving at time delay τ [40]. Up until now, there are many approaches to obtain the photon correlation g (2) (τ), typically including two-photon absorption (TPA) measurement [41,42], photon coincidence counting [43], time interval measurement of photon pairs [44]. Recently, the HBT experiment was explored to observe chaos from quantum dot lasers with external feedback [23]. However, research on photon correlation g (2) (τ) of high-dimensional chaotic waveforms is rare and there is still an obvious disagreement between experimental and theoretical g (2) (τ). The calculation of g (2) (τ) from the photon pair time interval distribution provides a good way to measure the photon correlation of pseudo-thermal light with microsecond coherence time [44]. But For chaotic laser the coherence time is much shorter than that of pseudo-thermal light, and the resolution time must be shorter than the coherence time of the laser in the measurement. Although the shorter coherence time does not affect the bunching effect or g (2) (0) of chaotic laser [24], that makes the measurement of g (2) (τ)(τ≠0)at very short timescales using HBT technique more difficult, owing to the limited response time of single-photon detectors [45]. It remains an important challenge to ravel the g (2) (τ) (τ≠0)of the chaotic laser at a high precision, whose coherence time is below 1 ns. Accordingly, high precision and ultrashort resolution time are required to acquire an accurate g (2) (τ) of the chaotic laser. That is, it is potentially useful to extract higher order coherence and achieve desired laser source for quantum imaging and secure communication.
In this paper, we theoretically and experimentally investigate the second order photon correlation g (2) (τ) of a chaotic optical-feedback semiconductor laser. The g (2) (τ) is precisely measured using self-convolution HBT detection at tens of picoseconds resolution time. A different high order correction of g (2) (τ) is analyzed and confirmed experimentally, which has a low relative error in wide range of delay time. It shows a good agreement between experimental results and theoretical analysis. We also measure the bunching effect and coherence time of chaotic laser using the precise photon correlation technique. This technique, avoiding the photon overlapping, can give a g (2) (τ) with a high accuracy. To the best of our knowledge, the accurate measurement of g (2) (τ) for the chaotic laser has not been investigated and reported. In view of this demonstration, we present first some highlights of precise photon correlation measurement that are necessary for a better understanding of quantum statistics of the chaotic laser. The demonstration well reveals photon correlation g (2) (τ) of chaotic laser and provide a way of studying chaos with quantum optics technique.

High Order Correction of g (2) (τ)
Theoretically, second order photon correlation of g (2) (τ) can be obtained from an ideal photon pair time interval distribution P1(τ).Using the self-convolution method, one can obtain any desired high order n, and the higher n of g (2) n (τ) is, the more accurate g (2) n (τ) tends to the ideal g (2) (τ). 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 (τ) to the ninth order but 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 D1(τ). Furthermore, g (2) (τ) can be calculated from the self-convolution of D1(τ).
The second order photon correlation g (2) (τ) have a proportional relation to G(τ),as follows: whereI is the average photon counting rate per time bin of the light field. G(τ) is the histogram of photons at delay time τ between two photon detection events. The relation between G(τ) and P1(τ) is given by, where P1(τ) is an ideal photon pair time interval distribution of light field which can be obtain based on HBT experiment, and Pn(τ) is nth order self-convolution of P1(τ) .When P1(τ) is less than one, the sum of Pn(τ) is convergence [40], then we can obtain where L denotes the Laplace transformation, and L -1 denotes the Inverse Laplace transformation.
When the above theory is applied to Lorentzian chaotic laser field, we can get the relation between G(τ) and Pn(τ) of a chaotic laser. The first order correlation of Lorentzian chaotic laser is as follows: The relation between g (2) (τ) and g (1) (τ) of Lorentzian chaotic laser is: Using Equations (4) and (5), we obtain: According to Equations (1) and (6),we obtain: The relation between g (2) (τ)and Pn(τ) is shown as follows.
P2(τ) is the self-convolution of P1(τ) Pn(τ) is the convolution of P1(τ) and Pn-1(τ) The integration upper bound of Equation (10) should be replaced by the maximum time interval τ practically. Now the new equation is Using Equation (3), (8), (9), (10), and (11), we obtain all of the self-convolution of P1(τ) Inserting Equation (7) to Equation (13) we can get different Pn(τ) . The form of Pn(τ) can be obtained by numerical self-convolution. The sum of Pn(τ) is G(τ), and in theory g (2) n (τ) is comparable to g (2) (τ)for sufficiently high n. In fact, with the increase of n, g (2) n (τ) is closer to ideal g (2) (τ). Using Equation (14) and increasing the order of n, we can obtain high order g (2) n (τ). Considering the realistic experiment condition and the data-processing ability, the maximum order of n we take is nine. The theoretical high order correction of g (2) (τ) is given above, which can help us to know the influences of the experimental parameters. Here, the direct self-convolution method is used to get g (2) n (τ) from experimental data. In that case, P1(τ) is related to the experimentally measured photon pair time interval distribution D1(τ). Dn(τ) is nth order self-convolution of D1(τ). Experimental results of g (2) n (τ) can be obtained from D1(τ) [44], and the relation between G(τ) and Dn(τ) is Thus, when we obtain the D1(τ), the high order correction g (2) n (τ) can be deduced from the experimental photon pair time interval distribution as follows The above analysis basically solves the high order correction g (2) n (τ)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 coherence time of chaotic laser. In addition, high order correction of g (2) (τ) for coherent light can be achieved and the g (2) (τ) is perfectly equal to one.
Using the Equation (14), g (2) n (τ) is calculated to ninth order, and higher order terms than ninth can be omitted. Relative error δ varying with the delay time τ at the correction order of nine is calculated as: (2) (2)

Experiment Setup
The experimental setup is shown in Figure 1, represents the aspheric lens collimators, and F is an optical filter used to filter out the background noise. The chaotic laser is divided into two equal intensity beams whose intensity are measured by the detectors SPD1 and SPD2. One can adjust the mean photon intensity of the light through the VA2.After the above steps, the photon pair time interval distribution can be attained.

Experimental Results
The chaotic laser is firstly attenuated by a variable attenuator and then passes through the HBT setup. In the photon detection system, an internal clock triggers two channel gates simultaneously. When a photon was detected on one channel, the arriving time is recorded. During the same clock period, a subsequent photon was received from another channel and then the time interval was measured. The desired distribution was obtained with many records, and if the detection quantum efficiency is higher, the better the photon pair distribution is close to the real light source distribution. Otherwise, the single photon detector would mistake dark noise for photon signals. Moreover, as the incident photon number increases, the noise level would be higher due to the after pulsing effect. In that case, the time interval distribution of photon pairs is also affected by noise. When the coherence time of light source is short, high resolution time is required in the detection. Besides, the unbalance of the two light intensity after the BS has an adverse effect on the acquired distribution. It was difficult to obtain an accurate time interval distribution of photon pairs with a very low quantum efficiency. In our experiment, the detection quantum efficiency is 25%. We investigate different average photon intensity and coherence time affect the accuracy of different order correction. We use the relative error to compare different high order correction with the ideal second order photon correlation. According to Equation (2), we calculateP9(τ) with high order terms and omitted the terms higher than ninth order. Likewise, we take photon pair time interval distribution D1(τ) and then convolve D1(τ)to D9(τ). The terms higher than ninth order is also omitted. Using Equation (17) we obtain different high order correction of g (2) (τ) with experiment data. The influence of different average photon intensity and the coherence time are investigated theoretically. At 1.5 times the threshold current (J=1.5Jth) and 25 °C temperature (T=25°C), central wavelength was stabilized near 1548 nm. We adjust the attenuator VA1 and polarization controller to accurately control the optical feedback strength. With the increase of the feedback strength, the laser experienced a transition from the period-1, period-2, to the steady chaos oscillation. Among them, we select three typical states, including period-2 (weak chaos) with the feedback strength η of 2.66%, the intermediate chaotic state (chaos) with η of 8.87%, and steady chaotic oscillation state (Strong chaos) with η of 30.31%. Figure 2a shows the three typical frequency spectrums of the chaotic laser. To quantify the bandwidth of the chaotic laser, we used the definition that is expounded as the frequency spectrum region the DC and the frequency where 80% of the energy is contained within [46]. According to the 80% bandwidth definition, the bandwidth of chaotic laser was4.98 GHz,9.84 GHz, and 11.71GHz, respectively. Figure 2b is the optical spectrum of the chaotic laser. Environmental changes slightly influence the optical feedback strength and the coherence length [47]. Based on the repeated measurements we obtained the range of coherent time variation. Figure 3 is the three corresponding time series of the chaotic laser.    The bandwidth of the chaotic laser is in the order of GHz and we obtained the coherence time of chaotic laser through 3dB linewidth spectrum. Considering that the ninth order correction of the second order photon correlation g (2) (τ) was close enough to theoretical limit, we experimentally took the ninth order correction within 10 ns and theoretically employ the same order fitting. The experimental photon correlation g (2) (τ) are fitted by ideal expressions, as shown in Figure 4. For photon-bunching chaotic light, the g (2) (τ) can be written as g (2) (τ)=1+bexp(-2τ/τc) (b: bunching amplitude, τc: coherence time) [23]. Figure 4 shows the experimental and theoretical fitting results for weak chaos (b=0.479, τc=0.768 ns), chaos (b=0.524, τc=0.651 ns), and strong chaos (b=0.626, τc=0.535 ns).

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 (~0.5 ns) of the chaotic laser, resulting in a little fluctuation of measured g (2) (τ). In Figure 5(a), the experimental results of g (2) n (τ) within 100 ns delay time is shown and the magenta curve represent the original photon pair time interval distribution. The original experimental data is the same as those used in Fig 4b. The bottom-up colored curves indicate the increasing order corrections of second order photon correlation. The orange curve indicate is the third order correction of g (2) (τ), and the others are fifth, seventh, ninth order correction of g (2) (τ). For an accurate measurement of photon correlation, a very low photon flux rate I was required to ensure [39].
The courting rate of the SPD was controlled below 0.

Relative Error of g
(2)

n (τ)With Mean Photon Intensity and Coherence Time of Chaotic Laser
The coherence time of our experiment is below 1 ns and we can set the maximum coherence time in the theoretical analysis. Following this g (2) (τ) is 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 is 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) (τ), it cannot provide sufficient information and accuracy according to Equation (2). For the ninth order correction, there was almost no difference between g 9 (τ) and g (2) (τ) and the loss information can be ignored. Figure 6 shows relative error of g (2) 9 (τ)for photon intensity changes form 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 40ns 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 Figure6, 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. Following this, we theoretically analyzed the coherence time from 0.3 ns to 0.7 ns under the condition that the photon intensity is near 4×10 7 photon/sec. Figure 7 shows the relative error as functions of the coherence time τc and the delay time τ. The coherence time τc varies from 0.3 ns to 0.7 ns and the delay time τ is within 100 ns. In this case, corresponding to our experimental condition, the relative error is not exceed 5‰ within 50 ns delay time. It is worth noting that long τc leads to big relative error, but the change of relative error is subtle. The relative error caused by the coherence time is smaller than that of the photon intensity. We compare the relative error caused by the above two factors (photon intensity and coherence time).The yellow dashed line in Figure 7 indicates the case that the coherence time is 0.5 ns, which corresponds to the experiment condition. For the same delay time, the relative error caused by coherence time was lower than that caused by photon intensity. Thus, high accuracy g (2) (τ) requires well controlling the photon intensity [24].
From the above discussion, the high order correction of second order photon correlation was affected by the variations of the mean photon intensity and coherence time of the laser, and we analyzed the relative error caused by the two factors respectively. The relative error from incident photon intensity was larger than that from coherence time. In Figure 7, the dashed line on the error surface was under the condition that the intensity was 0.04 photons/ns and τc was 0.5 ns, which corresponds to the experimental condition. In our experiment, the maximum relative error in ninth order correction of g (2) (τ) does not exceed 5‰ within 50 ns delay time. The relative errors caused by the photon intensity and coherence time retained the uncertainty ±0.01 photon/ns and±0.2 ns respectively, and the overall error within 50 ns delay time did not exceed 1% in our condition.

Conclusions
In conclusion, we precisely measured the second order photon correlation g (2) (τ) of a chaotic semiconductor laser using self-convolution HBT interferometer. Based on the theoretical analysis, the ninth order self-convolution correction was sufficient to obtain experimentally the accurate g (2) (τ) from the photon pair time interval distribution. The experimental results were in good agreement with the theory. The relative error caused by coherence time and mean photon intensity was analyzed, which was no more than 5‰ within 50 ns delay time. In comparison with the traditional HBT measurement, this technique, which does not require high intensity and long optical or electric delay, is more useful for a weak light source, such as atomic fluorescence and single photon emission, whose quantum correlation is difficult to be detected. It is demonstrated that this technique provides a new way to measure high order quantum coherence precisely and will bridge the gap between nonlinear optics of chaotic lasers and quantum physics.
Author Contributions: X.G. and Y.G. designed the whole work and wrote the manuscript; X.G. carried out the theoretical calculations and analyzed the data; Y.G. supervised the experiments; C.C. and T.L. contributed to the experiment and data processing; X.F. analyzed the data and edited the manuscript. All authors discussed the results at all stages. All authors have read and approved the final manuscript.