Orbital angular momentum light frequency conversion and interference with quasi-phase matching crystals

Light with helical phase structures, carrying quantized orbital angular momentum (OAM), has many applications in both classical and quantum optics, such as high-capacity optical communications and quantum information processing. Frequency conversion is a basic technique to expand the frequency range of fundamental light. The frequency conversion of OAM-carrying light gives rise to new physics and applications such as up-conversion detection of images and high dimensional OAM entanglements. Quasi-phase matching (QPM) nonlinear crystals are good candidates for frequency conversion, particularly for their high-valued effective nonlinear coefficients and no walk-off effect. Here we report the first experimental second-harmonic generation (SHG) of OAM light with a QPM crystal, where a UV light with OAM of 100 is generated. OAM conservation is verified using a specially designed interferometer. With a pump beam carrying an OAM superposition of opposite sign, we observed interesting interference phenomena in the SHG light; specifically, a photonics gear-like structure is obtained that gives direct evidence of OAM conservation, which will be very useful for ultra-sensitive angular measurements. We also develop a theory to reveal the underlying physics of the phenomena. The methods and theoretical analysis shown here are also applicable to other frequency conversion processes, such as sum frequency generation and difference-frequency generation, and may also be generalized to the quantum regime for single photons.

Allen [20,21] has demonstrated the OAM transformation and conservation in frequency conversion in LBO crystal. Zeilinger's group [12] has realized high-dimensional OAM entanglement in the spontaneous parametric down-conversion processes. In all these nonlinear interaction processes, the total OAM conservation of light plays a very important role. The frequency conversion of OAM lights will be very useful in up-conversing detection of images [26] and generating of OAM light from a fundamental OAM light at special wavelengths (in the UV or mid-infrared frequency domains), which are hard to produce them with traditional method. For nonlinear processes with crystals, the benefits from quasi-phase matching (QPM) when compared with birefringence phase matching make QPM crystals good candidates for frequency conversion of OAM light, particularly for their high-valued effective nonlinear coefficients and no walk-off effect. Then some important questions are coming naturally: can we use QPM crystals for nonlinear frequency conversion of OAM light? Whether the total OAM of light is conserved in such nonlinear processes? Is frequency conversion of OAM superposition state possible? So far, no experimental work has been reported on such frequency conversion processes, although one theoretical study has appeared on the process of sum frequency generation (SFG) and second-harmonic generation (SHG) [27].
In this work, the previous posed questions are answered, we report the first experimental generation of OAM-carrying UV light in SHG with a QPM type-I PPKTP crystal. We demonstrate the conservation of OAM in the SHG process, which concurs with the theory of Ref. 27. Moreover, we observe a very interesting interference phenomenon by transforming the pump light into a hyper-superposition of polarization and OAM states. We directly see a photonic gear-like structure that has never before been observed or discussed in three-wave mixing processes. This phenomenon can be regarded as direct evidence of OAM conservation. The photonic gear can be rotated by rotating the pump-beam polarization, an effect that can be used for ultra-sensitive angular measurements. These observations can be well explained by the theory we have developed. The method we demonstrate here provides a new way to generate OAM light via frequency conversion in QPM crystals. Moreover, because of its low diffraction, UV light can enhance the resolution of OAM light-based imaging. Using the SHG process in OAM light-based ultra-sensitive angular measurements [18], resolutions can be further enhanced by a factor of 2. Our approach may also be used in sum frequency generation (SFG) or difference-frequency generation (DFG) [28] at the single-photon level. This will be very useful for quantum information processing using the OAM degrees of freedom of photons.
We first demonstrate OAM conservation in the SHG process. Figure 1 shows the different blocks used in our experiments; blocks a, b, and e are used in the conservation demonstration. Block-a is used to generate OAM light with the proper polarization using vortex phase plates (VPPs, from RPC photonics). Block-b performs frequency conversion and comprises two lenses (both have the same focus length of 125mm), a type-I PPKTP crystal, and a UV filter used to remove the pump light. The 1 mm×2 mm×10 mm PPKTP crystal, supplied by Raicol Crystals, was designed for SHG of wavelengths from 795 nm to 397.5 nm. Both end faces are anti-reflection coated for these two wavelengths. The measured nonlinear conversion efficiency for the PPKTP crystal is 1%/W for a Gaussian pump mode.
In our experiments, the pump power was 10 mW, which produces an SHG light power of around 1 μW.
The laser light we used was from a continuous wave Ti: sapphire laser (Coherent, MBR 110, less than 100 KHz line width when locked). The measured phase matching temperature of the crystal is 64.3°C.
The temperature of the crystal was controlled with a semiconductor Peltier cooler with stability of ±2 mK. Block-e is a specially designed balanced interferometer for generating light in a superposition of  Block-e is a specially designed balanced interferometer used to determine the OAM value of the input light. Block-f has the same function as block-c, but uses an SLM instead of a VPP. When we use pump-beam light with a hyper-superposition of polarization and OAM states for SHG, a photonic gear-like structure is obtained. Before showing the experimental results, we first give a detailed theoretical description. We use quantum mechanics to describe the transformation of light in block-c or -f, and a configuration similar to that presented in Refs. 17, 29, and 30 is used in our experiment. We assume that the input beam of the interferometer is in a Gaussian mode and is polarized in the horizontal direction. The input state can be expressed as where H denotes the polarization degree of freedom and 0 represents the OAM degrees of freedom. After passing through block-c (or block-f), the light is transformed into the state ( ) [(cos (2 ) sin (2 ) ) (sin (2 ) cos (2 ) ) where θ and δ are the angles of the fast axis of the half-wave plate (HWP) with respect to the vertical axis at the respective input and output ports of the block, and l is the OAM quantum number imprinted on the two counter-propagated beams in the interferometer. The output SHG light is in the form (see Supplementary Information for details) where Γ is a constant of renormalization. This expression shows that the output of the SHG light is a superposition of OAM states of 2l , 0, and 2l − that depends on the angle of δ . We now focus on the case 8 π δ = ; apart from a relative phase of 8θ , the first two terms have the same amplitude. As mentioned before, an interference pattern with 4l maxima is generated in the intensity distribution of the outer ring, giving direct evidence of OAM conservation in the SHG process. More interesting is that the interference pattern can be rotated if the phase θ is changed, indicating that the total phase of the pump beam is preserved in the SHG process. This behaviour is similar to a mechanical gearwhen θ changes by 4 π , the pattern rotates through angle 2l π and can be exploited for ultra-sensitive measurements of angles. Furthermore, by changing δ, we can switch easily between states 2l , 2l LG . For small OAM, the diffraction of the LG mode is blurred and hard to see; only a dim point can be distinguished at the centre, and hence we cannot observe the multi-ring structure. By rotating the angle of the HWP in the input port of the interferometer in block-c, a rotation in the output image is observed. We also find that the image of the SHG light is clearer than the input; this is because waves of shorter wavelength are diffracted less. ; the first state is the same as that prepared using VPPs, the second is an asymmetrical state. Using this configuration, the two counter-propagated beams have the same optical length with an intrinsically stable phase between them. The results are shown in Figure 4. The first image in each row is the phase diagram of the SLM for generating LG modes with specific l value; the other images are similarly arranged as in For each l , the number of maxima in the intensity profile of the outer ring is the same as theoretically predicted. For large l , there is an additional SHG light in the central region (rows e, f) arising from limitations in creating the mode at the SLM (which are arising from high-order LG mode with the same OAM and unmodulated light, respectively). There would be no such artefact if high-quality VPPs were used (see row b in Figure 3 for comparison). In row e, the OAM of UV is 100, corresponding to 200 maxima in its intensity profile. We cannot increase the OAM further because the SLM cannot operate at high powers; also, our CCD camera has a limited resolution.
For the asymmetrical state 7 LG , 16 0 LG − , and 1 7 LG − ; the interference pattern of the pump has 15 maxima, whereas the SHG light has 30 maxima. The pattern is not sufficiently clear as the LG modes with different absolute values of l have different diffraction properties. Hence the two modes do not completely overlap in the far-field. The first image in each row is the phase diagram of SLM for generating a specific OAM-carrying light. The second and fourth images are the respective interference patterns for the pump light, projected onto the diagonal polarization direction, and the SHG light, directly observed after block-b using CCD camera. The third and fifth images are the corresponding theoretical patterns.
In summary, two experiments using the type-I QPM PPKTP crystal have been conducted to investigate OAM transformation and conservation in the SHG process. In the first of the two, we verified that OAM is conserved in the SHG by directing the pump and SHG OAM light into a specially designed balanced interferometer. The conservation law is confirmed by counting the maxima in the interference intensity profile. As the QPM crystal has a high-valued effective nonlinear coefficient and no walk-off effect, it provides a new method to generate OAM light by frequency conversion in QPM crystals. The image resolution depends on the wavelength of light used; shorter wavelengths yield better image resolutions. UV OAM light would be suitable for OAM light-based phase imaging. In the second of the experiments, we observed a very interesting interference phenomenon when pumping the PPKTP crystal with a superposition of two OAM states of opposite sign. The output SHG light intensity profile depended on the polarization of the pump light. A photonics gear-like structure is observed that can be rotated when the pump polarization is rotated. This effect can be used for remote sensing, OAM light-based ultra-sensitive angular measurements, and detection of spinning objects [31]. This interference effect can also be used for optical switching between different SHG patterns generated by controlling the polarization of the pump beam. We also gave analytical expressions for propagation of the SHG light for the tight focus approximation. All experimental phenomena can be well explained within the theory we have developed. For SFG and DFG conversions, the method is not limited to just the classical regime, and can be extended into the quantum regime for single photons. We use the language of quantum mechanics to describe the transformation of light via blocks c or f. We use a configuration similar to that described in Refs. 17, 29, and 30. We assume a Gaussian spatial mode polarized in the horizontal direction for the input beam of the interferometer. The input state can be expressed in the tensor product form in which the first state, here H , gives the polarization degrees of freedom and the second, here 0 , gives the OAM degrees of freedom. The function of the half-and quarter-wave plates is to apply a unitary rotation to the polarization degrees of freedom. We use the Jones calculus notation, with convention The functions of the quarter-and half-wave plates, whose fast axes are at angles ϕ and θ with respect to the vertical axis, are given by the respective 2×2 matrices ( ) ( ) cos(2 ) sin(2 ) 1 , sin (2 ) cos (2 )  2 cos (2 ) sin(2 ) 1 , sin (2 ) cos (2 ) After passing through the plates, the polarization of the beam becomes ( ) ( ) ( ) ( )