Interference between two independent multi-temporal-mode thermal fields

We construct a general theoretical model for analyzing the intensity correlation of the field formed by mixing two independent multi-temporal-mode thermal fields. In the model, we use the intensity correlation function $g^{\left(2\right)}$ to characterize the mode property of the mixed thermal field. We find that $g^{\left(2\right)}$ of the mixed field is always less than that of the individual thermal field with less average mode number unless the two thermal fields are identical in mode property. The amount of drop in $g^{\left(2\right)}$ of the interference field depends on the relative overlap between the mode structures of two thermal fields and their relative strength. We successfully derive the analytical expressions of the upper bound and lower limit for $g^{\left(2\right)}$ of the interference field. Moreover, we verify the theoretical analysis by performing a series of experiments when the mode structures of two independent thermal fields are identical, orthogonal, and partially overlapped, respectively. The experimental results agree with theoretical predictions. Our investigation is useful for analyzing the signals carried by the intensity correlation of thermal fields.

Figure 1

Schematic of Hanbury Brown–Twiss (HBT) interferometer for intensity correlation measurements of (a) a thermal field and (b) an interference field formed by mixing two individual thermal fields, respectively. BS, 50:50 beam splitter; D, detector.

Figure 2

First three Hermite-Gaussian modes of a temporal mode basis in (a) frequency domain and (b) time domain, respectively.

Figure 3

Experimental setup for verifying the interference formed by mixing two multi-mode thermal fields $E_1$ and $E_2$. ${\mathrm{P}}_1-{\mathrm{P}}_2$, pump; ${\mathrm{DSF}}_1-{\mathrm{DSF}}_2$, dispersion shifted fiber; ${\mathrm{F}}_1-{\mathrm{F}}_2$, filter; ${\mathrm{TF}}_1-{\mathrm{TF}}_2$, tunable filter; ${\mathrm{FPC}}_1-{\mathrm{FPC}}_2$, fiber polarization controller; ${\mathrm{FPBS}}_1-{\mathrm{FPBS}}_2$, fiber polarization beam splitter; ${\mathrm{PBS}}_1-{\mathrm{PBS}}_2$, polarization beam splitter; ${\mathrm{VOA}}_1-{\mathrm{VOA}}_2$, variable optical attenuator; HWP, half wave plate; ${\mathrm{BS}}_1-{\mathrm{BS}}_3$, 50:50 beam splitter; DL, delay line; ${\mathrm{SPD}}_1-{\mathrm{SPD}}_2$, single photon detector. The solid lines and dotted lines in the scheme respectively denote the optical fiber propagation and free space propagation.

Figure 4

Intensity correlation function of the interference field $g_m^{\left(2\right)}$ for two thermal fields $E_1$ and $E_2$ with different relative intensity coefficient $\mathcal{R}$. The results in (a) and (b) are obtained by fixing the mode number of $E_1$ at $M_1=1.25$ but setting the mode number of $E_2$ to $M_2=1.25$ and $M_2=2.5$, respectively. The error bars of data are within the size of data points. The solid circles are obtained for $E_1$ and $E_2$ with the mode profiles satisfying the upper bound conditions, $cos\theta =1$ and $K_{ik}={\delta }_{ik}$, while others (squares, diamonds, and triangles) are obtained when the mode profiles of $E_1$ and $E_2$ satisfy the lower limit conditions, $cos\theta =0$ or $K_{ik}=0$. The thick and thin curves are the results calculated by substituting mode numbers of $E_1$ and $E_2$ into upper bound and lower limit of $g_m^{\left(2\right)}$ in Eqs. (29) and (32), respectively.

Figure 5

Intensity correlation function of the interference field $g_m^{\left(2\right)}$ measured by (a) varying the angle of the polarization $\theta$ between linearly polarized $E_1$ and $E_2$ and by (b) varying the delay $\tau$ between $E_1$ and $E_2$. In the measurement, $M=M_1=M_2=1.25(g_m^{\left(2\right)}=g_1^{\left(2\right)}=g_2^{\left(2\right)}=1.8)$ and $\mathcal{R}=0.5$. The error bars of data are within the size of data points. The solid curves in (a) and (b) are obtained by substituting experimental parameters in the formulas $g_m^{\left(2\right)}=1+\frac{1}{2M}(1+{cos}^2\theta )$ and $g_m^{\left(2\right)}=1+\frac{1}{2M}\{1+exp[-{\tau }^2{\sigma }^2{(g^{\left(2\right)}-1)}^2/2]\}$, respectively.

Figure 6

Intensity correlation function $g_m^{\left(2\right)}$ of the interference field formed by mixing two thermal fields $E_1$ and $E_2$ varies with different relative intensity coefficient $\mathcal{R}$. The squares and diamonds are obtained for $E_1$ and $E_2$ with mode numbers of (a) $M_1=M_2=1.67$ and (b) $M_1=1.25,M_2=1.67$, respectively. The error bars of data are within the size of data points. The dashed curves are fittings of Eq. (35) with $\mathcal{V}=0.62$ for (a) and 0.82 for (b). The thick and thin curves are the theoretical predictions of upper bound and lower limit, calculated by substituting mode numbers of $E_1$ and $E_2$ into Eqs. (29) and (32), respectively. In the measurement, $E_1$ and $E_2$ are originated from two independent spontaneous Raman scattering processes with different basis of temporal modes; the conditions $0<K_{i,k}<1$ and $cos\theta =1, \tau =0$ are satisfied.

Figure 7

Intensity correlation function of interference field $g_m^{\left(2\right)}$ versus the relative intensity coefficient $\mathcal{R}$ when the mode numbers of thermal field $E_1$ are fixed at $M_1=1.05$ but the mode number of thermal field $E_2$ is (a) $M_2=1.11$, (b) $M_2=1.67$, and (c) $M_2=2.5$, respectively. The error bars of data are within the size of data points. The dashed curves are obtained by fitting the data (solid circles) to Eq. (35) (dashed curves) with the best-fit value $\mathcal{V}$ = 0.93 for (a), 0.62 for (b), and 0.48 for (c), respectively. The thick and thin curves, representing the upper bound and lower limit of the interference effect, are calculated by substituting mode numbers of $E_1$ and $E_2$ into Eqs. (29) and (32), respectively. In the measurement, $E_1$ and $E_2$ are originated from the nonlinear processes of spontaneous four wave mixing and spontaneous Raman scattering, respectively, and the conditions $0<K_{ik}<1$ and $cos\theta =1,\tau =0$ are satisfied.