Orbital angular momentum (OAM) of light continues to gain interest in structured light applications, where beams with OAM have a structured phase that can be exploited to encode and decode information. The Poynting vector of OAM beams spiral along the propagation direction [1] with a helical wave-front, $\exp (i\ell \theta)$, where $\ell$ is the OAM charge number, and $\theta$ is the azimuthal coordinate. Integer values of OAM form a Hilbert space, with fractional OAM largely considered a composition of integer OAM and not an eigenstate of light [2]. The interaction of beams with OAM in a nonlinear medium can be further exploited to create new frequencies of light with new OAM charge numbers.

When exploiting nonlinear processes with structured light, the OAM conservation law is a useful tool to understand the OAM interactions as a result of the nonlinear process. Examples including second-harmonic generation (SHG) of Laguerre–Gaussian (LG) modes [3,4], OAM modes with Gaussian distribution [5], and perfect vortices [6] have been investigated. For multiple OAM interactions, four-wave mixing based on mode overlap theory [7] and Gaussian beam propagation theory [8] have been explored. Additionally, the SHG and sum frequency generation (SFG) of OAM were investigated by embedding a twisted pattern into a nonlinear crystal [9]. The interference phenomena of superimposed OAM in SHG was shown in [10], and high-harmonic generation (HHG) of vortices were produced in the extreme ultraviolet in [11] with a controllable OAM. Polarization-controlled switching of OAM was demonstrated in [12], and the propagation property of OAM was also studied in nonlinear conversion [13]. In [14], the SFG of half-integer OAM was studied, and fractional-phase-plate generated OAM SHG conversion was studied using spectrum decomposition theory in [15]. The SHG of half-integer OAM modes was studied in [16], the SHG of fractional OAM generated by shifting an integer phase plate was analyzed [17]. Most confirmed the SHG of half-integer OAM generates integer OAM, making it difficult to realize fractional states in nonlinear generated beams with OAM. Also, power densities between different OAM components need to be evaluated, since a shift occurs from the predicted integer OAM, due to the spatial overlap of the coherent combinations of OAM [18].

This Letter demonstrates the nonlinear frequency conversion of an asymmetric perfect vortex (APV) with fractional OAM. This asymmetry in the amplitude avoids the phase discontinuity and results in a fractional OAM primarily related to the azimuthal phase gradient. This eliminates the need for the fundamental beam to be decomposed into an integer OAM spectrum. By controlling the amplitude and phase distribution, we illustrate the fractional OAM-carrying beam with the APV yields a good one-to-one correspondence with the OAM mapping. Thus, different fractional OAM values carried by the APV can be considered discernable. Multiple OAM interactions are also studied using a real-time OAM cross-correlator method based on an acousto-optic deflector (AOD) and log-polar optics to verify the interaction.

In order to create these APV beams of fractional order at the fundamental, we use the near-field profile for the temporal OAM generation higher-order Bessel–Gaussian beams integrated in time (HOBBIT) system (OAM transmitter) and the real-time OAM cross-correlator (OAM receiver) based on a single-pixel detector HOBBIT system for verification of the OAM spectrum [19,20]. The transmit and receive systems are based on log-polar optics and the acousto-optic (AO) effect [19,20], as shown in Fig. 1. By utilizing an AOD to generate the linear phase tilt that is mapped to OAM, every OAM mode is frequency tagged due to the Doppler shift from the AO interaction. This enables one to exploit the interaction between different OAM modes, since their Doppler frequency shifts tag the fundamental OAM values and can be further utilized to analyze the SHG fields. The asymmetric perfect optical vortex beam generated from this system can be described as follows:

 

Fig. 1. Experimental setup. (a) OAM transmitter. (b) The real-time OAM cross correlator.

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The beam generated by the transmitter is shown in Fig. 2(a). The APV is the result of amplitude and phase modulation, which leads to no discontinuity in the beam profile. As a result, the global OAM is only related to the azimuthal phase gradient. In Fig. 2(b), the OAM mapping curve is measured using the fundamental HOBBIT beams at 1064 nm [21], which is the far-field of the APV. It is important to note that the fundamental global OAM value is preserved when propagating to the far-field. Because of the asymmetry property of both the near-field and far-field, our beams are structurally stable upon propagation [18], compared to structurally stable fractional beams based on mode superposition [23]. A 20 mm long periodically poled lithium niobate (PPLN) nonlinear crystal (Covesion MSHG1064-1.0-20) with a grating period of 6.93 µm was used as the nonlinear medium in the type zero quasi-phase-matching condition. The receiver is a real-time OAM cross correlator working at 532 nm, as shown in Fig. 1(b), which is the reverse process of the HOBBIT OAM generator. The log-polar optics transform the spiral OAM phase to a linear phase, and then this linear phase will correlate with the phase grating of the receiver AOD driven by a linear RF signal on the receive side. The OAM receiver provides the OAM mapping and frequency/phase spectrogram of the beat frequency [20], which can be used for the verification of the SHG field.

 

Fig. 2. (a) APV-carrying OAM charge 0.5. (b) One-to-one fundamental fractional OAM mapping [21] measured through the cylindrical lens method [22].

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The coupled wave equation for SHG is as follows [24]:

Mode overlap, power density differences, and phase matching all need to be considered to determine the final global OAM [18]. An advantage of APVs is that beams with fractional OAM all have the same amplitude distribution, so that in a nonlinear process involving multiple OAM interactions, the overlap of the beam profiles is maximized. Since the field distributions are identical, the power density of each OAM can be controlled, which means that the power spectrum is known. Another factor in the nonlinear interaction will be the phase-matching differences between each component. It is well known that the Poynting vector of OAM beams spiral along the beam axis while propagating. For an APV, every OAM beam has the same intensity distribution, then the phase-matching conditions are different for each mode. When the beam radius is large enough, the change to $\Delta k$ will be negligible. When this condition is satisfied for a beam including $n$ OAM components, Eq. (4) can be expressed as follows:

Because the length of the nonlinear crystal is relatively long, a beam diameter of 300 µm is used to ensure a minimal change of the propagation properties of the APV. With this beam size, the phase-matching condition can be considered the same for OAM in the range of ${-}{10}$ to ${+}{10}$. This is seen by calculating the efficiency for an OAM value of ${+}{10}$ at the perfect phase-matching condition for OAM 0 using Eq. (5), which results in ${{\rm sinc}^2}({\Delta kL/2}) = 0.9534$. Experimental results are shown in Fig. 3. Figure 3(a) shows the intensity distributions of the fundamental and SHG fields captured by a beam profile camera. The incident fundamental OAM is ${-}{1}$, ${-}{0.5}$, 0, ${+}{0.5}$, and ${+}{1}$, respectively. The measured SHG OAM in Fig. 3(b) shows a good correlation with theory, some residual peaks are noted due to the AOD alignment and the higher-order diffraction. Figure 3(c) shows the SHG fractional one-to-one OAM mapping with the fundamental OAM from ${-}{1}$ to ${+}{1}$ with an increment of 0.1. From results shown in Fig. 3, we can see the results correlate well with theory using the undepleted-pump approximation for the OAM interaction. The APV not only carries fractional OAM, but also simplifies the conversion process for any nonlinear process involving OAM conversion and interaction. The OAM interaction in a nonlinear process is interesting when multiple OAM beams are involved. Using the undepleted-pump approximation, substituting Eq. (1) into Eq. (6) results in the generated SHG field described by Eq. (7):

 

Fig. 3. (a) Intensity distribution of fundamental and SHG field. (b) SHG OAM measurements resulting from the incident fundamental OAM of ${-}{1}$, ${-}{0.5}$, 0, ${+}{0.5}$, and ${+}{1}$. (c) The OAM mapping of the SHG field for the fundamental OAM from ${-}{1}$ to ${+}{1}$ with increments of 0.1.

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Since each fundamental OAM is related to a specific frequency as a result of the AO Doppler shift, the SHG field of the generated OAM will contain a summation of frequencies. This property can be utilized to analyze the OAM in the SHG field. The analysis process can be described by Eqs. (8) and (9) [25]. In Eq. (8), $z(t)$ is a signal carrying a summation of sinusoidal waves, in which each frequency component has a specific amplitude and phase. Equation (9) shows how to evaluate the information carried by beat frequencies:

An example of using this beat frequency information to analyze the OAM in the SHG field is shown in Fig. 4. The fundamental beam is a combination of three OAM modes, $\pm0.5$ and 0. Considering all the interactions between the different OAM beams, according to Eq. (7), the generated SHG field will include $\pm1$, $\pm0.5$, and 0. In the current transmitter, the fundamental OAM modes that vary by one unit will have a frequency difference of 5.8 MHz. The SHG OAM modes interfere, producing beat frequencies that are dependent on the frequency difference of the interacting modes. The beat frequencies generated by the interacting SHG OAM modes are given in Fig. 4(b). Interacting OAM components having the same change in OAM will have the same beat frequency, so for clarity, Fig. 4(b) gives an example of interacting OAM modes for each beat frequency generated. Compared with the fundamental OAM modes, the SHG OAM modes that vary by one unit will also have a frequency difference of 5.8 MHz. By setting the initial phase values of fundamental OAM ${-}{0.5}$ for ${\phi _n} \in [{0,\frac{\pi}{4},\frac{\pi}{2},\pi ,\frac{{3\pi}}{4}}]$, there will be interference in the SHG field, which will give insight to the multiple OAM interaction. Figure 4(a) is the measured OAM spectrogram with the initial phase of the ${-}{0.5}$ OAM beam changing in 2 ms increments. The mapping shows the interference between different OAMs. By applying a fast Fourier transform (FFT) to each column of Fig. 4(a), the beating information is revealed in the frequency spectra. The spectrum reveals four main peaks in the frequency field shown in Fig. 4(c), which represents 2.9 MHz, 5.8 MHz, 8.7 MHz, and 11.6 MHz.

 

Fig. 4. Experimental results of multiple OAM interaction. (a) OAM spectrogram of the measured SHG field. (b) The relationship between OAM step and beat frequency. (c) Frequency spectrogram of the SHG field where the magnitude of the spectrum indicates the power ratio of each component.

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A frequency and phase spectrogram of each beat frequency component is shown in Fig. 5(a). The magnitude change at different time periods shows the constructive and destructive interference caused by the initial phase setting. From the phase spectrogram, the phase gradient shows a regular change. Using Eqs. (8) and (9), the final phases for 2.9 MHz, 5.8 MHz, 8.7 MHz, and 11.6 MHz will be $0.5{\phi _n}$, ${\phi _n}$, $1.5{\phi _n}$, and $2{\phi _n}$. Extracting the phase information in Fig. 5(b), the curves demonstrate the phase change with time. At 2.9 MHz and 5.8 MHz, the curve shows very accurate phase steps, which correspond to $0.5{\phi _n}$ and ${\phi _n}$. Frequencies of 8.7 and 11.6 MHz have a reduced signal level, which reduces the phase measurement accuracy. However, these measurements demonstrate a way to frequency tag the OAM and their interactions in the nonlinear process.

 

Fig. 5. (a) Detailed frequency and phase spectrograms of Fig. 4(c). (b) Extracted phase information at beat frequency 2.9 MHz, 5.8 MHz, 8.7 MHz, and 11.6 MHz.

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In conclusion, we have shown the fractional OAM conversion in a nonlinear process using APV beams. Since the global OAM of the light field is impacted by both the phase and amplitude distribution, the APVs have a good one-to-one OAM mapping. Compared to conventional fractional OAM beams described using an LG basis, the APV has a much simpler phase distribution, which leads to benefits in nonlinear conversion. The SHG theory of the APV is studied, and the OAM in the SHG field has a good one-to-one correspondence and agrees with the OAM conservation law. Although the asymmetry change caused by the nonlinear conversion will increase the uncertainty or spread in OAM, which means a wider spiral spectrum [26], the global OAM is conserved. The multiple OAM interactions of the APVs are also tagged with their Doppler frequency shifts of the individual OAM beams, with the OAM receiver detecting the beating information of any two OAM beams to verify the multiple OAM interaction theory. In addition, the OAM conversion in the nonlinear processes offers a method to create new frequencies that carry higher-order OAM, which can be used in a variety of applications for imaging and sensing based on integer and/or fractional states of OAM.