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Abstract
We demonstrated a method to achieve the two-photon subwavelength effect of true broadband chaotic light in polarization-selective Michelson interferometer based on two-photon absorption detection. To our knowledge, it is the first time that this effect has been observed with broadband chaotic light. In theory, the two-photon polarization coherence matrix and probability amplitudes matrix are combined to develop polarized two-photon interference terms, which explains the experimental results well. To make better use of this interferometer to produce the subwavelength effect, we also make a series of error analyses to find out the relationship between the visibility and the degree of polarization error. Our experimental and theoretical results contribute to the understanding of the two-photon subwavelength interference, which shed light on the development of the two-photon interference theory of vector light field based on quantum mechanics. The characteristic of the two-photon subwavelength effect have significant applications in temporal ghost imaging, such as it helps to improve the resolution of temporal objects.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
How to break through the Rayleigh limit has always attracted people’s attention, mainly because it plays an important role in optical lithography and biological imaging [1,2]. Jacobson et al. theoretically proposed the concept of Photonic de Broglie waves in 1995 and pointed out that the corresponding de Broglie wavelength is $\lambda /N$ for the compound system of $N$ photon entanglement, where $\lambda$ is the wavelength of a single-photon [3]. And then Fonseca et al. experimentally measured the de Broglie wavelength of two-photon entangled states which generated by type-II Spontaneous Parametric Down-Conversion (SPDC) for the first time [4]. Later, Boto and D’Angelo et al. proposed quantum lithography that the resolution could beat the classical diffraction limit by a factor of 2 via the two-photon entangled subwavelength interference [5,6]. However, it is difficult to generate multi-photon entangled states using the nonlinear interaction between light and matter since the efficiency is low, which is not conducive to practical applications [7–9]. Subsequently, theoretical and experimental studies have shown that thermal light plays a similar role to the quantum entanglement of two-photon state in subwavelength interference [10–13]. Two-photon subwavelength interference of pseudothermal light has been measured in the Hanbury Brown-Twiss (HBT) interferometer when the two detectors are scanned in the opposite directions, whereas it also makes the high-order correlation subwavelength interference of thermal light can not be directly applied to quantum lithography [10–16]. Later, the methods of reversing the wavefront and the design of new classical light sources or other ways have been proposed to overcome these defects [7–9,17,18].
Up to now, the reported light sources that can achieve two-photon subwavelength interference including entangled photon pairs [3–6,19], pseudothermal light [10–13], coherent light [17,18,20], and so on [21,22]. As far as we know, the two-photon subwavelength interference effect of true broadband chaotic light has never been observed, mainly because the two-photon subwavelength interference effect of chaotic light is difficult to detect due to its femtosecond scale coherence time. It was not until 2009 that Boitier et al. proposed to use a detector based on two-photon absorption (TPA) to measure the photon bunching effect [23]. The detection response rate makes it can obtain the signal on the order of a few femtoseconds [24,25]. In this way, many meaningful phenomena have been measured based on TPA detection, such as the extrabunching effect [26,27], ultrabroadband ghost imaging [28], ghost polarimetry [29–31], and superbunching effect of broadband chaotic light [32]. In fact, the counts detected by the TPA detector contain much different two-photon interference information, including the second-order interference term, the background term, the subwavelength interference term, and so on. Each of these terms contains corresponding interference information and is worth studying. Based on this, we hope to use a method to control the two-photon probability amplitudes terms to modulate the two-photon subwavelength interference phenomenon based on TPA detection.
In this paper, we proposed and demonstrated a method to achieve the temporal subwavelength effect of true broadband chaotic light in polarization-selective Michelson interferometer based on TPA detection. Also, we developed the polarized two-photon interference matrix by introducing the two-photon polarization coherence matrix in the quantum two-photon interference theory, which is in good agreement with the experimental results. It is different from the traditional method to achieve subwavelength interference in the transverse plane, our technique can allow us to measure the two-photon subwavelength interference of chaotic stationary light at the timescale of a few femtoseconds, which may find potential applications in the optical polarimeter [29] and optical interferometer [33], especially can improve the resolution of objects in temporal ghost imaging [34].
The remaining parts of the paper are organized as follows. In Sec. 2, we will combine the quantum two-photon interference theory with the polarization theory to interpret the observed two-photon subwavelength interference. In Sec. 3, we will manipulate the two-photon interference phenomena by placing three polarizers at different positions in the Michelson interferometer. The discussions about the physics of the subwavelength interference and its analysis of experimental error are in Sec. 4. The conclusion is in Sec. 5.
2. Theory
Generally, when the quantum two-photon interference theory was employed to study the second-order coherence effect of thermal light, the usual practice was to simplify the light field as a scalar field [32,35,36], and ignore the polarization properties of light produced by the vectorial electromagnetic field. Once polarizers are inserted in the optical path, the input light will change from unpolarized light to polarized light. Then the employed theory we study will change from scalar to vector, so that the corresponding two-photon interference terms will change accordingly and result in different interference phenomena [37], such as the two-photon subwavelength interference effect.
It is well known that the bunching effect of chaotic stationary light in the Michelson interferometer can be explained by quantum two-photon interference theory [32,38]. When we develop vector two-photon interference theory, polarization is treated as a new dimension in the phase space of the system, just like spatial, temporal, and spectral coherence properties. Coherent superposition of two-photon probability amplitudes happens only in the same coherent mode of polarization also like those happen in spatial and temporal domain [13,35,39]. In addition, the polarization can act as a switch to manipulate the quantum two-photon interference that happens in the Michelson interferometer. Many new effects can be observed by manipulating polarization setups and the subwavelength effect is one of them. To develop the theory of two-photon subwavelength interference, we need to matrix the two-photon interference terms and then bring the two-photon polarization coherence matrix into the related TPA detection probability amplitudes matrix, and finally achieve the two-photon subwavelength effect. The two-photon interference matrix based on TPA detection modulated by arbitrary polarizers can be expressed as [37]
where ${\textbf {M}_\textbf {TPA}}$ is the two-photon probability amplitudes matrix, ${\textbf {J}_\textbf {P}}$ is the polarization coefficient matrix of the two-photon coherence terms. Also, ${\textbf {J}_\textbf {P}}$ determines whether the TPA detection probability amplitudes terms are present, just like a switch.Firstly, we briefly review the quantum two-photon interference theory based on the Michelson interferometer. The two-photon bunching effect is caused by the superposition of the probability amplitudes corresponding to different and indistinguishable paths triggering the TPA detector. We define $a$ and $b$ as two photons from the chaotic light. Channel 1 and channel 2 can be defined as the different paths that the photons are reflected from ${\rm M_1}$ and ${\rm M_2}$, respectively. There are four different but indistinguishable ways for photons $a$ and $b$ to trigger the TPA detector, corresponding to four two-photon probability amplitudes which are $A_{a 1 b 1}, A_{a 1 b 2}, A_{a 2 b 1}$, and $A_{a 2 b 2}$. The probability of a TPA detection event happening is
There are 16 probability amplitudes multiplied by each other, which we can view as the sum of all the terms in a $4\times 4$ matrix. Therefore, the probability amplitudes matrix based on TPA detection can be expressed as
When we add three polarizers with different polarization angles to the optical path, where ${\rm P_1}$ is at the input of the interferometer and set to $0^{\circ }$ polarization (horizontal), ${\rm P_2}$ and ${\rm P_3}$ are set to $45^{\circ }$ with respect to the horizontal polarization at the two arms of the interferometer, the employed theory would be transformed from a scalar theory to a vector theory. The polarization coefficient matrix of the two-photon coherence terms can be expressed as [36,37]
To show more directly which two-photon probability amplitude terms are modulated by the corresponding two-photon polarization coherence matrix, we take the absolute value of ${\textbf {J}_{\textbf {P}}}$ and its matrix visualization is shown in Fig. 1(a). $i$, $j$ represent the columns and rows of the matrix, respectively.
Fig. 1. (a) shows the matrix visualization of the absolute value of ${\textbf {J}_{\textbf {P}}}$, the values represented by red, yellow and blue histogram are $\frac {3}{8}$, $\frac {1}{8}$ and $0$ respectively. The probability amplitudes above the histogram represent the two-photon interference terms in Eq. (3) that can be modulated by this polarization matrix. (b) shows the simulation results of a two-photon interferogram with three polarizers, which corresponds to Eq. (6).
According to the Feynman path integration theory [32,40], the probability amplitude can be expressed as $A_{aibj}=e^{i \varphi _{a}} K_{a i} e^{i \varphi _{b}} K_{b j}$, where $\varphi _{a}$ and $\varphi _{b}$ are the initial phase of photon $a$ and photon $b$, respectively [41]. $K_{a i}$, $K_{b j}$ are the Feynman propagators of different photons. Since the thermal light source has a certain spectral distribution, which is different from the single frequency of the laser. In this case, the Feynman propagator of the temporal correlation from the thermal light can be expressed as [42] $K\left (t_{1}-t_{2}\right )=\int _{\omega _{0}-\frac {1}{2} \Delta \omega }^{\omega _{0}+\frac {1}{2} \Delta \omega }f(\omega )e^{-i\omega \left (t_{1}-t_{2}\right )} d \omega$, where $f(\omega )$ is the spectral distribution function of light source that assumed the rectangular spectrum distribution, $\omega _{0}$ is the center frequency and the spectral bandwidth is $\Delta \omega$, which all parameters depend on the spectral characteristics of the light source we are using. $\tau =t_{1}-t_{2}$ means the time difference that also equals $\tau = Z/c$, where c is the speed of light and Z is the optical-path difference. Equation (3) can be simplified by the above formulas.
Substituting Eq. (3) and Eq. (4) into Eq. (1), the polarized two-photon interference matrix can be simplified as
After adding each term of the two-photon interference matrix ${\textbf {M}_{\textbf {P}}}$, the normalized two-photon interference phenomenon based on TPA detection controlled by polarizers is
The simulation results of Eq. (6) are shown in Fig. 1(b), it is actually made up of three different two-photon interference terms. 1 is the constant term corresponding to the solid black line in the inset of Fig. 1(b). $\frac {1}{2}\left [1\textrm { + }\operatorname {sinc}^{2}\left ( {\frac {1}{2}\Delta \omega \tau } \right )\right ]$ is the second-order coherence function corresponding to the solid red line in the inset of Fig. 1(b). $\frac {1}{2}\left [1\textrm { + }\cos \left ( {2{\omega _0}\tau } \right )\operatorname {sinc}^{2}\left ( {\frac {1}{2}\Delta \omega \tau } \right )\right ]$ is the two-photon subwavelength interference term corresponding to the solid green line in the inset of Fig. 1(b). Owing to the oscillation frequency at $2{\omega _0}$, it means the measured period will be half of the previous bunching effect.
3. Experiment
By placing three polarizers at different positions in the Michelson interferometer, the two-photon subwavelength interference phenomenon of broadband chaotic light can be observed. The experimental setup is shown schematically in Fig. 2. A continuous amplified spontaneous emission (ASE) incoherent light is used in the configuration, which is completely unpolarized light as like natural light. This source serves as a 1547 nm center wavelength whose bandwidth is 39 nm. A light that we analyze is coupled into a single-mode optical fiber (SMOF) and delivered to the Michelson interferometer which consists of two mirrors (${\rm M_1}$ and ${\rm M_2}$) and a beam splitter (BS). These polarizers ${\rm P_1}$, ${\rm P_2}$, ${\rm P_3}$ will be placed as shown in Fig. 2. ${\rm P_1}$ is at the input of the interferometer and set to $0^{\circ }$ polarization (horizontal). ${\rm P_2}$, ${\rm P_3}$ are set to $45^{\circ }$ with respect to the horizontal polarization, where are in front of ${\rm M_1}$, ${\rm M_2}$ respectively. The lens ${\rm L_1}$ and ${\rm L_2}$ are two convergent lenses with focal lengths of 10 mm and 25.4mm, respectively. The role of ${\rm L_2}$ is to focus the output beam of the interferometer into the semiconductor photomultiplier tube (PMT) (Hamamatsu H7421-50), allowing the detector to operate in a two-photon absorption regime as much as possible. The high-pass filter (HF) with a cutoff wavelength of 1300 nm is used to eliminate the single-photon counting of the PMT. They are placed in front of the TPA detector successively.
Fig. 2. Experimental setup used to modulate the two-photon subwavelength interference based on the polarized Michelson interferometer. ${\rm M_1}$ and ${\rm M_2}$ are 1550 nm dielectric mirrors, where ${\rm M_1}$ is fixed on a precise motorized linear translation stage (MS). ${\rm P_1}$, ${\rm P_2}$, ${\rm P_3}$ are linear polarizers. SMOF is a single-mode optical fiber. PMT is a GaAs photomultiplier tube.
In order to make a comparison between the ordinary bunching effect and the two-photon subwavelength interference effect, we first measured its photon bunching effect. Firstly, no polarizers is placed at the Michelson interferometer. The second-order coherence function of the bunching effect is measured by scanning one arm of the interferometer-scanning ${\rm M_1}$ on a motor stage (MS). The dark counts are 120 per second in the measurement. The TPA counts are recorded versus the time delay difference between the two arms. The experimental results are shown in Fig. 3(a) by the green line, a fragment of which is shown in the inset for $\tau \in \left [ {\textrm {{ - }}15,15} \right ]$fs. The measured oscillation period is ${\tau _1} = 5.25$fs, and its corresponding optical-path difference is $\rm {Z_1} = 1.575\mu m$. The corresponding intensity correlation function is found by numerically low-pass filtering the oscillating curves and shown as the red solid line in Fig. 3(a). The degree of second-order coherence ${g^{(2)}}(0) = 1.92 \pm 0.02$ could be obtained after normalization. Since there are no polarizers, the observed two-photon bunching effect of chaotic light in the Michelson interferometer [32] comes from the coherent interference of these four probability amplitudes in Eq. (1).
Fig. 3. The measured two-photon absorption counts and the related intensity correlation functions under different conditions. (a) The measurement results of the bunching effect on the basis of no polarizers at the interferometer. (b) The measurement results of two-photon subwavelength interference when three polarizers (${\rm P_1}$, ${\rm P_2}$, ${\rm P_3}$) with different polarization angles are placed in the Michelson interferometer are shown as in Fig. 2. The green line is a graph of all counting results of the TPA detector. The red lines represent the normalized second-order correlation function. The insets show an enlarged view in both interferograms, while both insets are equipped with the scale of the time difference and the optical-path difference.
In the next step, ${\rm P_1}$, ${\rm P_2}$, ${\rm P_3}$ are inserted into the setup to constitute the polarization-selective Michelson interferometer as shown in Fig. 2. In this way, the ASE source will become horizontally linearly polarized light after passing through ${\rm P_1}$. Then the collimated beam from ${\rm L_1}$ is split at the BS, giving two beams taht propagate through ${\rm P_2}$ and ${\rm P_3}$ in two different arms of the interferometer, where ${\rm P_2}$ and ${\rm P_3}$ are all set to $45^{\circ }$ with respect to the horizontal polarization in the clockwise direction. Although ${\rm P_3}$ is set to $45^{\circ }$, the beam reflected from ${\rm M_2}$ converged on BS and then reflected out of the Michelson interferometer, its polarization angle becomes $135^{\circ }$. Therefore, the two light beams from ${\rm M_1}$ and ${\rm M_2}$ are actually perpendicular to each other when they propagate through BS, which are $45^{\circ }$ and $135^{\circ }$ linearly polarized light respectively.
Two beams are combined at the output of the interferometer to trigger the TPA detector and the measurement results are shown in Fig. 3(b). The green line is the measured TPA counts by the polarization-selective Michelson interferometer. The red solid line is the result of removing the high-frequency oscillation term by numerical filtering, which corresponds to the degree of second-order coherence ${g^{(2)}}(0) = 1.89 \pm 0.01$. Most notably, the inserted details inside the interferogram have an oscillation period of ${\tau _2} = 2.75$fs, corresponding to the optical-path difference of $\rm {Z_2} = 0.825\mu m$. This is almost half of the period measured in Fig. 3(a), which means that two-photon subwavelength interference can be observed by choosing the two-photon probability amplitudes terms based on the polarization-selective Michelson interferometer.
In our experiment, there are some systematic errors, such as the position of BS relative to ${\rm M_1}$ and ${\rm M_2}$, or the adjustment of the angle of the polarizers, it is extremely easy to cause the visibility of the two-photon subwavelength interference to decrease. The visibility of the two-photon subwavelength interference is only 6.2%. Specific error analysis will be described in detail in Sec. 4.
4. Discussion
To further study the two-photon subwavelength interference effect of broadband chaotic light, we simulate and compare the two-photon interference patterns based on TPA detection in two cases as shown in Fig. 4. The red circles indicate the details of the TPA detection event when chaotic light goes through the Michelson interferometer with no polarizers, which corresponds to the result of Eq. (2). The period of the interference pattern is 5.2fs (optical-path difference equals ${\rm 1}.56\mu m$). The simulation also shows that the relative TPA counts will change from 8 to 0 when the phase changes from 0 to $\pi$. From these TPA counts, a visibility $V = \left ( {{C_{\max }} - {C_{\min }}} \right )/\left ( {{C_{\max }} + {C_{\min }}} \right )$ with ${V} = 100\%$ is obtained. By contrast, the green circles represent the details of the TPA detection event based on the polarization-selective Michelson interferometer, and it corresponds to the result of Eq. (6). It can be found that the interference period of 2.6fs (optical-path difference equals ${\rm 0}.78\mu m$) is half of the previous two-photon interference pattern, which is the two-photon subwavelength interference effect we observed. In addition, the visibility is reduced from 100% to 20%, and the TPA counts are reduced from 3 to 2 when the phase changes by half a period.
Fig. 4. Simulation results of two-photon interference with different polarization conditions. The red circles show the details of Eq. (2) without polarizers. The green circles show the details of Eq. (6) modulated by three polarizers, which corresponds to the two-photon subwavelength interference effect. The width of the red shadow and the green shadow represent their periods respectively.
Since there is a certain error between the measured two-photon subwavelength interference effect and the simulation results, we have done a set of error analyses under the same conditions to further study the source and magnitude of the error in the two-photon subwavelength interference phenomenon. The simulation results of polarization degrees with different errors are shown in Fig. 5(a), the red solid line represents the simulation results in the most ideal experimental conditions with the polarization error of $0^{\circ }$. Three polarizers (${\rm P_1}$, ${\rm P_2}$, ${\rm P_3}$) are placed in the experimental setup as shown in Fig. 2, in which the angle of polarization is set to $0^{\circ }$, $45^{\circ }$, and $135^{\circ }$, respectively. Therefore, the two-photon subwavelength interference effect can be observed, whose visibility reaches the maximum of 20%.
Fig. 5. (a) shows the simulation results of the two-photon subwavelength interference effect under different polarization errors. The red square, orange circle, yellow upward triangle, green downward triangle, blue rhombus, and purple pentagram represent details of the TPA detection event when the polarization error is $2^{\circ }$, $4^{\circ }$, $6^{\circ }$, $8^{\circ }$, and $10^{\circ }$, respectively. (b) shows the visibility of the two-photon subwavelength interference effect under different polarization errors. The black circles are measured results and the red lines are fitted curves, the blue solid circle represents the polarization error and visibility of the two-photon subwavelength interference measured in Fig. 3(b).
However, it is obvious that the two-photon subwavelength interference effect is gradually disappearing, when we add a series of polarization errors into the ideal simulation system, such as $2^{\circ }$, $4^{\circ }$, $6^{\circ }$, $8^{\circ }$, and $10^{\circ }$. With the increase of the degree of polarization error, the counts of the red shadow in Fig. 5(a) will become higher and higher, which indicates that the visibility will decrease. While the green shadow of Fig. 5(a) will become lower and lower, indicating that more and more terms containing $\cos \left ( {{\omega _0}\tau } \right )$ are mixed into Eq. (6). This is exactly the reason why the phenomenon of subwavelength interference effect measured in Fig. 3(b) has a lower background than the simulated background of 2 in Fig. 1(b). In addition, we also calculated the visibility of the two-photon subwavelength interference effect under different polarization errors as shown in Fig. 5(b). When the polarization error is from $0^{\circ }$ to $10^{\circ }$, the visibility drops from a maximum of 0.20 to 0. The black circles are measured results and the red line is the fitting of the measured data. Since the visibility of the two-photon subwavelength interference measured in experiment is 6.2%, the polarization error is $4.860^{\circ } \pm 0.001^{\circ }$ according to the red fitted curve, as shown as the blue solid circles in Fig. 5(b).
Therefore, it can be seen that the insertion of polarizers with specific angles in the TPA-based Michelson interference can indeed achieve the two-photon subwavelength interference effect, the essence is to introduce the two-photon polarization coherence matrix into the two-photon probability amplitudes matrix. However, in the actual experiment, we should pay special attention to the systematic error to avoid the error is too large to observe the two-photon subwavelength interference effect.
5. Conclusion
In summary, we have developed a polarization-selective Michelson interferometer to explore the two-photon subwavelength interference effect of broadband chaotic light based on the TPA detection at a femtosecond timescale. The subwavelength interference effect is achieved by manipulating the quantum two-photon interference effect, in which we use polarizers as switches to control the coefficient of different two-photon probability amplitudes. By combining the two-photon polarization coherence matrix with the TPA detection probability amplitudes matrix, the polarized two-photon interference matrix of the subwavelength interference effect is presented. The theoretical simulation results are in agreement with the experimental results. It may achieve super-resolution of phase in temporal ghost imaging. To better apply the two-photon subwavelength effect, we also make a series of error analyses to explain the visibility difference between the experimental results and the simulation results, and finally obtained that the polarization error of the system was $4.860^{\circ } \pm 0.001^{\circ }$. The discussions of the two-photon subwavelength interference effect, in both the theory and experiment, are helpful to understand the second-order coherence of the vector light field. We anticipate that these results will be an important tool for studying the fundamental and applied aspects of optical coherence, such as the ghost polarimetry and temporal ghost imaging.
Funding
Shaanxi Key Research and Development Project (Grant No. 2019ZDLGY09-09); National Natural Science Foundation of China (Grant No. 61901353); Key Innovation Team of Shaanxi Province (Grant No. 2018TD-024); 111 Project (Grant No. B14040).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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