The linear Doppler effect describes a phenomenon that electromagnetic waves or sound waves reflected from a moving object will cause a frequency shift, which has been extensively utilized to measure the linear velocity of an object [1]. Analogously, when light is reflected or emitted from a rotating object, it will undergo a frequency shift, which is the rotational version of Doppler effect [2]. The rotational Doppler effect, associated with photon spin angular momentum, was originally observed and analyzed by Garetz et al., who used rotating half-wave plates (HWPs) of angular velocity to imprint a frequency shift to circularly polarized light [3]. Subsequently, the rotational Doppler effect was also manifested by using millimeter waves, in which the amount of rotational Doppler shift is related to the orbital angular momentum (OAM) [4,5]. It was Lavery and coworkers that demonstrated the rotational Doppler effect in the visible light region, which enabled the measurement of angular velocity with the light beam scattered from the rough surface of a spinning object [6]. From the applied point of view, the rotational Doppler effect has seen a wide range of applications for angle remote sensing, from the microscopic world to the aerospace field [7]. Through the coherent interaction between OAM beams and rubidium atoms at room temperature, Barreiro et al. reported on the first spectroscopic observation of rotational Doppler shift [8]. Luo et al. theoretically predicted that when an OAM beam interacts with a left-handed material, the rotational Doppler effect will induce a transfer of angular momentum from the material to the beam [9]. Korech et al. demonstrated a new method to control and detect the spinning molecule microscopically based on the rotational Doppler effect [10]. Zhou et al. demonstrated the optical rotational Doppler effect via theoretical analysis and experimental verification, and they developed the OAM complex spectrum analyzer based on this effect [11,12]. In a city environment, we built a 120 m free-space link between the rooftops of two buildings to detect the angular velocity of rotating objects from the observed rotational frequency shifts [13]. Besides, rotational Doppler effect was exploited to deduce the angular acceleration rate of a spinning object [14]. Also, Qiu et al. investigated the influence of the lateral misalignment and the tilted angle on the measurement of rotational Doppler effect [15,16]. Recently, we explored the quantum correlations of spatially entangled photons for angle remote sensing in a nonlocal manner [17].

It is worth noting that infrared remote sensing may be more advantageous in certain applications, such as infrared monitoring in the military realm [18], non-destructive and reagent-free biological detection [19–21], remote sensing of urban climate [22], and ocean remote sensing [23,24]. However, suffering from exorbitant prices, relatively low quantum efficiency, severe dark current, and requiring additional cryogenic cooling, the development of infrared detection is seriously affected [25]. Moreover, it is difficult to achieve simple and efficient image detection in the mid-infrared regime with existing detector technologies [26,27]. To circumvent this constraint, ultrasensitive infrared detection can be expected with nonlinear frequency conversion from the infrared light field to the visible spectral region by leveraging the high sensitivity of visible single-photon counting modules [25,28–32]. This strategy has been demonstrated to build all-optical upconversion imaging systems for few-photon-level full two-dimensional infrared imaging [33]. Here, via second-harmonic generation (SHG), we extend this scheme to measure the rotational Doppler shifts from the infrared detected light to visible with a high-efficiency single-photon detector. Actually, a clever scheme has been devised by Li and coworkers to report the rotational Doppler effect in nonlinear optics, in which the frequency shifts were imparted by the rotating nonlinear crystal itself [34,35]. Instead, we rotate one fundamental wave (FW) by illuminating it onto a spinning object while preparing the other FW to carry the desired OAM superpositions and thus enabling the detection of frequency shifts and angular velocity of rotating objects from SHG light.

The Laguerre–Gaussian (LG) modes, as one kind of typical OAM eigenmode, are conceived as a set of orthogonal and complete basis vectors in high-dimensional Hilbert space. In the cylindrical coordinate $(\rho ,\varphi )$, a standard LG mode at the beam waist plane $z=0$ can be written as $${\mathrm{LG}}_p^{\ell }(\rho ,\varphi )=R_p^{\ell }(\rho )\exp (i\ell \varphi )$$(1)and $$R_p^{\ell }(\rho )=C_{p,\ell }{\left(\frac{\sqrt{2}\rho }{\omega }\right)}^{|\ell |}\exp \left(\frac{-{\rho }^2}{{\omega }^2}\right)L_p^{|\ell |}\left(\frac{2{\rho }^2}{{\omega }^2}\right),$$(2)where $p$ and $\ell$ are the radial and azimuthal mode indices, $\omega$ is the beam waist, $C_{p,\ell }$ is the normalized constant, and $L_p^{|\ell |}(·)$ is the generalized Laguerre polynomial. According to the concept of digital spiral imaging [36], we can describe the transmission function or the rough surface of any rotating object $E_1(\rho ,\varphi )$ in terms of LG modes, namely, $E_1(\rho ,\varphi )={\sum }_{\ell ,p}{A}_{\ell ,p}{\mathrm{LG}}_p^{\ell }(\rho ,\varphi )$, where the complex amplitude, $A_{\ell ,p}=\iint [{\mathrm{LG}}_p^{\ell }(\rho ,\varphi ){]}^{*}\cdot E_1(\rho ,\varphi )\rho \mathrm{d}\rho \mathrm{d}\varphi$. Here we define the normalization coefficients $a_{\ell }=A_{\ell ,p=0}/A_{\ell =0,p=0}$ for the OAM modes. For an easy visualization, we plot in Fig. 1 the OAM mode intensity spectra ${|a_{\ell }|}^2$ for three typical objects of different rotational symmetries, i.e., a three-leaf clover, a hexagonal snowflake, and a cartoon pattern. It can be observed that the principal components of LG modes are distributed over those modes with the azimuthal indices of $\ell =0,\pm 3,\pm \mathrm{6,}\dots$ for the clover in Fig. 1(a) and over $\ell =0,\pm 6,\pm \mathrm{12,}\dots$ for the snowflake in Fig. 1(b), which reveals the threefold and sixfold rotational symmetries, respectively. Another object in Fig. 1(c), the cartoon pattern, contains a variety of LG mode components ranging from $\ell =0$ to $\ell =\pm 6$ and thus reflects its own structural property of low symmetry.

The mathematical picture of eigenmode decomposition offers us a more intuitive understanding on the physical origin of rotational Doppler effect. Assume that the rotating object has a constant angular velocity $\mathrm{\Omega }$ around its own axis; as a result, we can describe the transmission function of the object as $$\begin{gathered} E_1(\rho ,\varphi )\to E_1(\rho ,\varphi +\mathrm{\Omega }t) \\ =\sum _{\ell ,p}{A}_{\ell ,p}{R}_p^{\ell }(\rho )\exp (i\ell \varphi )\exp (i\ell \mathrm{\Omega }t). \end{gathered}$$(3)

It can be seen clearly from Eq. (3) that, as a result of rotation, an additional time-dependent and OAM-related phase variation of $\exp (i\ell \mathrm{\Omega }t)$ has been individually induced to each OAM component, which is just equivalent to a frequency shift of $\ell \mathrm{\Omega }$. This result has been well known and extensively reported in the realm of linear optics [2,4–16]. However, to the best of our knowledge, only a little attention has been paid to extend the rotational Doppler effect to nonlinear optical realm [34,35]. Such an extension is crucial for realizing high-efficiency infrared detection via frequency upconversion. Instead of rotating the nonlinear crystal as in Refs. [34,35], we directly mix the infrared FW bearing the rotational Doppler signal of Eq. (3) with another FW carrying desired OAM superpositions. Then, via SHG, we can realize the transfer of rotational Doppler signals from the infrared to the visible region. Under the paraxial approximation and the slowly varying amplitude approximation, we can describe the generation of SHG light field by the following wave-coupling equation [37]: $$\frac{\mathrm{d}{E}_{\mathrm{SHG}}(\rho ,\varphi ,t)}{\mathrm{d}z}=\frac{i2\pi {\mathrm{d}}_{\mathrm{eff}}}{{\lambda }_{\mathrm{vis}}{n}_{\mathrm{vis}}}{E}_1(\rho ,\varphi +\mathrm{\Omega }t)E_2(\rho ,\varphi ),$$(4)where $d_{\mathrm{eff}}$ is the effective nonlinear coefficient, $z$ is the propagation distance, and ${\lambda }_{\mathrm{vis}}$ and $n_{\mathrm{vis}}$ are the visible light wavelength and the crystal refractive index, respectively. Here we use type II phase-matching condition to do SHG, in which two FWs should be of orthogonal horizontal and vertical polarizations such that self-interaction of each FW can be well suppressed. Then, in the case of small signal approximation, the SHG light field can be simply expressed as $$E_{\mathrm{SHG}}(\rho ,\varphi ,t)\propto E_1(\rho ,\varphi +\mathrm{\Omega }t)E_2(\rho ,\varphi ).$$(5)

In order to extract the rotational Doppler signals of the infrared light $E_1(\rho ,\varphi +\mathrm{\Omega }t)$, we prepare the other FW as two specific OAM mode superpositions, namely, $E_2(\rho ,\varphi )=[\exp (i\mathrm{m}\varphi )+\exp (-i\mathrm{m}\varphi )]/\sqrt{2}$. This process is just equivalent to using an OAM filter of $\ell =\pm m$ to perform the detection of the intensity modulation frequency $f_{\mathrm{mod}}=m\mathrm{\Omega }/\pi$ and thus infer the angular velocity $\mathrm{\Omega }$ [6]. Unlike the linear optical case, the intensity modulation frequency $f_{\mathrm{mod}}$, is measured from the SHG light as a result of the transfer of rotational Doppler shift in the process of SHG. Besides, it is worth mentioning that the possible rotational symmetry of the detected objects can also be inferred from the measured Doppler signals, as its intensity is also determined by ${|a_{\ell }|}^2$, being related to the rotational symmetry of the object. We demonstrate these key results of frequency upconversion and symmetry-determined rotational Doppler effect in the following experimental section.

In our experimental implementation, as shown in Fig. 2, the horizontally polarized light beam derived from a near-infrared 1064 nm laser (Cobolt, Rumba) is expanded and collimated by a telescope, and then it is directed to illuminate a computer-controlled spatial light modulator (SLM, Hamamatsu, X10468-7). The SLM is divided into two regions, each of which has a pixel array of $400\times 600$. We simulate the rotating objects by preparing time-varying holographic gratings in region I and display the desired OAM superpositions in region II. As the light beams reflected from these two regions are nearly collinear, we employ an additional D-shaped mirror to separate them spatially. Then, after probing the rotating objects, one infrared FW is imparted with the rotational Doppler signals $E_1(\rho ,\varphi +\mathrm{\Omega }t)$, in which each OAM mode individually experiences a time-dependent phase shift of $\exp (i\ell \mathrm{\Omega }t)$ as manifested by Eq. (3). In order to extract the angular velocity of the rotating object, we prepare the other FW in the conjugated OAM superpositions $E_2(\rho ,\varphi )=[\exp (i\mathit{m}\varphi )+\exp (-i\mathit{m}\varphi )]/\sqrt{2}$. Besides, as the potassium titanyl phosphate (KTP) crystal is cut for type II phase matching, we build a polarization-dependent interferometer to recombine these two FWs. For this, we insert an additional HWP with its fast axis orienting at 45° in one arm to rotate the horizontal polarization of the OAM beam into the vertical one. After a polarizing beam splitter (PBS), the recombined beam is incident on a $5\mathrm{mm}\times 5\mathrm{mm}\times 5\mathrm{mm}$ KTP crystal to participate in SHG process. Note that a $4f$ system consisting of two lenses ($f_1=750\mathrm{mm}$ and $f_2=100\mathrm{mm}$) with an iris in each focal plane is nested to filter out the first-order diffraction, which is equivalently imaged onto the facet of KTP crystal. The visible SHG light is then obtained after a narrow filter centered at 532 nm. Thus, according to Eqs. (3) and (5), we can write the SHG light field in terms of OAM modes as $$\begin{gathered} E_{\mathrm{SHG}}(\rho ,\varphi ,t)\propto \sum _{\ell ,p}{A}_{\ell ,p}{R}_p^{\ell }(\rho ) \\ \times \{\exp [i(\ell +m)\varphi ]+\exp [i(\ell -m)\varphi ]\}\exp (i\ell \mathrm{\Omega }t). \end{gathered}$$(6)

Subsequently, we use another $4f$ system ($f_3=150\mathrm{mm}$ and $f_4=10\mathrm{mm}$) to collect and couple the SHG light into a single-mode fiber (SMF) that is connected to a single-photon counting module (SPCM). Note that SMF is acting as a zero OAM mode filter such that only those photons satisfying $\ell =\pm m$ can be detected by SPCM. In the backprojection picture, we can equivalently consider that a fundamental Gaussian mode is emitted from the SMF and then illuminates the crystal after being expanded by the $4f$ system [38]. We estimate that the beam waist of the expanded Gaussian mode is $\sim 1.0\mathrm{mm}$, being sufficiently larger than the spot of SHG light $\sim 0.25\mathrm{mm}$ at the output plane of the crystal. Thus, the combination of SMF and SPCM can be regarded as a bucket detector to collect all the rotational Doppler signals. Then we know from Eq. (6) the time-dependent intensity modulation can be written as $I_m(t)\propto {|a_m|}^2[\cos (2m\mathrm{\Omega }t)+1]$. The transfer of rotational Doppler frequency shift is manifested by the OAM-dependent intensity of SHG light with a modulation frequency $f_{\mathrm{mod}}=m\mathrm{\Omega }/\pi$ after a straightforward Fourier transform ${\tilde{I}}_m(f)=F\{I_m(t)\}$.

In our first set of experiments, we simulate a rotating object of three-leaf clover by using a computer-generation hologram in SLM. For the rotating clover, we set $\mathrm{\Omega }=1\text{deg/s}$ and $2\text{deg/s}$, respectively. For each case, we measure the rotational frequency shifts by scanning the OAM superpositions from $m=\pm 1$ to $m=\pm 6$. We illustrate in Fig. 3 the experimental results, in which Figs. 3(a) and 3(b) are the photon counts of SHG light recorded by SPCM while Figs. 3(c) and 3(d) are the power spectra of rotational Doppler signals after fast Fourier transform (FFT). In each case, the cosine oscillations of single-photon events can be seen clearly when OAM superpositions $m=\pm 3$ and $m=\pm 6$ are used. In contrast, the single-photon signals for other OAM modes are too weak to identify. As mentioned above, this could be easily understood from the threefold rotational symmetry of the three-leaf clover; see Fig. 1(a). Thus, we use the peak power signals in the recorded Doppler spectra to deduce the angular velocities of the rotating clover. For Fig. 3(c), it is observed that $f=0.0171\mathrm{Hz}$ at $m=\pm 3$ and $f=0.0341\mathrm{Hz}$ at $m=\pm 6$. Then, according to the relation $f_{\mathrm{mod}}=m\mathrm{\Omega }/\pi$, we have $\mathrm{\Omega }=1.0260\text{deg/s}$ and $\mathrm{\Omega }=1.0230\text{deg/s}$, with the relative error both smaller than 2.60%. For Fig. 3(d), we observe $f=0.0342\mathrm{Hz}$ at $m=\pm 3$ and $f=0.0683\mathrm{Hz}$ at $m=\pm 6$. Therefore, we have $\mathrm{\Omega }=2.0520\text{deg/s}$ and $\mathrm{\Omega }=2.0490\text{deg/s}$, with the relative errors both smaller than 2.60%.

Without loss of generality, we further consider other objects of more complex structures, such as a snowflake and a cartoon pattern. In our experiment, both patterns are set to rotate at the same constant angular velocity of 4 deg/s. For the snowflake of Figs. 4(a) and 4(c), we can see that the dominant Doppler signals of SHG light appear at the measured OAM superpositions of $m=\pm 6$ and $m=\pm 12$, in which the cosine oscillations of single-photon events can be seen clearly. This is again due to the obvious sixfold rotational symmetry of the snowflake as an echo of Fig. 1(b). We observe that $f=0.1318\mathrm{Hz}$ at $m=\pm 6$ and $f=0.2637\mathrm{Hz}$ at $m=\pm 12$, which give rise to the measured angular velocities of $\mathrm{\Omega }=3.9540\text{deg/s}$ and $\mathrm{\Omega }=3.9555\text{deg/s}$, respectively, with the relative errors both smaller than 1.15%. In contrast, for the cartoon pattern that has a relatively low symmetry, we plot the rotational frequency shifts by scanning the OAM superpositions from $m=\pm 1$ to $m=\pm 6$; see Figs. 4(b) and 4(d). We observe the obvious cosine signals for $m=\pm 1$, $\pm 2$, $\pm 3$, and $\pm 6$. As mentioned in Fig. 1(c), these results also manifest the fact that the cartoon pattern contains a variety of LG mode components, as a result from its own structural property of low symmetry. Similarly, we use the peak signals in the power spectra to deduce the angular velocities of the rotating cartoon pattern. We observe from Fig. 4(d) that $f=0.0220\mathrm{Hz}$ at $m=\pm 1$, $f=0.0440\mathrm{Hz}$ at $m=\pm 2$, $f=0.0660\mathrm{Hz}$ at $m=\pm 3$, and $f=0.1319\mathrm{Hz}$ at $m=\pm 6$, respectively. Therefore, we have $\mathrm{\Omega }=(3.9592\pm 0.0015)\text{deg/s}$, which again confirms the effectiveness of our scheme.

In conclusion, based on frequency upconversion, we have developed a new scheme for detecting the angular velocities of rotating objects via SHG. We have used one infrared FW to probe a rotating object and amplified the tiny angular velocity by the OAM numbers of the other FW as a result of transferring the rotational frequency shifts from the infrared to the visible region at low photon-count rates. Besides, the measured power spectra of Doppler signals also reflect the geometric symmetry of the rotating objects. Also, the periodically poled crystals, e.g., periodically-poled potassium titanyl phosphate (PPKTP), can be a better candidate to enable strong photon–photon interaction and to realize high-efficiency frequency conversion [39]. Thus, infrared monitoring and remote sensing even at the few-photon level will become more feasible in the future studies.

Acknowledgment. The authors thank Dr. Tong Liu from Space Engineering University for helpful technical discussions.