Frequency conversion of structured light

Coherent frequency conversion of structured light, i.e. the ability to manipulate the carrier frequency of a wave front without distorting its spatial phase and intensity profile, provides the opportunity for numerous novel applications in photonic technology and fundamental science. In particular, frequency conversion of spatial modes carrying orbital angular momentum can be exploited in sub-wavelength resolution nano-optics and coherent imaging at a wavelength different from that used to illuminate an object. Moreover, coherent frequency conversion will be crucial for interfacing information stored in the high-dimensional spatial structure of single and entangled photons with various constituents of quantum networks. In this work, we demonstrate frequency conversion of structured light from the near infrared (803 nm) to the visible (527 nm). The conversion scheme is based on sum-frequency generation in a periodically poled lithium niobate crystal pumped with a 1540-nm Gaussian beam. We observe frequency-converted fields that exhibit a high degree of similarity with the input field and verify the coherence of the frequency-conversion process via mode projection measurements with a phase mask and a single-mode fiber. Our results demonstrate the suitability of exploiting the technique for applications in quantum information processing and coherent imaging.

atomic vapor 46 . A crucial requirement for the exploitation of frequency conversion in applications of spatially encoded quantum information is that the conversion process be coherent, i.e. it must be possible to convert superpositions of modes with different OAM. This issue can be addressed by frequency-converting Hermite-Gauss (HG) beams, which can be represented as a superposition of Laguerre-Gauss (LG l ) modes with different OAM content l 47 , and employing a coherent detection scheme. Frequency conversion of beams in a superposition of different l has very recently been demonstrated by other groups for SHG 48 and SFG 49 .
Here, we report on the frequency conversion of HG spatial modes, i.e. superpositions of modes with different OAM content, from the near infrared (803 nm) to the visible spectral range (527 nm). The motivation is two-fold: 1) this demonstrates the suitability of frequency-converting spatially encoded states for applications in quantum information processing; and, 2) it shows how spatial information stored in the phase of a structured light beam can be converted to other wavelengths, which may extend phase-sensitive imaging and spectroscopy into new wavelength regions.
The optical frequency converter presented here is based on SFG with a strong Gaussian pump beam. In order to efficiently frequency up-convert the phase information stored in the wave front with minimum aberrations, we employ two key technological developments. First, the use of a periodically poled nonlinear crystal, which allows the interaction to be tailored to a non-critical phase-matching configuration without spatial walk-off, while at the same time using the largest nonlinear coefficient of the material. Second, the availability of powerful off-the-shelf fibered pump sources near 1550 nm based on erbium-doped fiber amplifiers. In the succeeding part of the article, we show that the 527-nm sum-frequency field exhibits the expected characteristic far-field intensity profile of the 803-nm input beams, and verify the preservation of coherence between different OAM values using a mode projection technique with a phase mask and a single-mode fiber.

Theoretical Background
In order to get an intuitive understanding of SFG with structured light, let us briefly discuss the expected spatial distribution of the output field using standard quantum optics formalism 50 . We assume that the pump is a monochromatic coherent field with a frequency ω p , and that the signal field is a single-photon, or weak coherent state with a frequency ω s . Our objective is to transfer the spatial structure of the signal field to the sum frequency mode, which is initially in its vacuum state. The signal field is described by its mode function in transverse momentum space ( ) u q s . We assume that the pump is Gaussian beam with its beam waist w located at the center of the nonlinear crystal. We consider only the prototypical experimental scenario in which fields propagate along a principal crystallographic axis of a periodically poled nonlinear crystal of length L and nonlinear coefficient χ ( ) 2 . Due to energy and momentum conservation, the conditions denotes the respective longitudinal wave vector for = q 0, and Λ is the poling period of the nonlinear crystal. Under these conditions the spatial state of the sum-frequency field can be expressed as: where vac denotes the initial vacuum state of the sum-frequency mode, and ( ) + a q is the creation operator for a photon with a transverse momentum component q. The spatial mode of the sum-frequency wave reads: . Due to symmetry, the total OAM is conserved for collinear SFG ( + = ) l l l p s SFG . Also, since the Gaussian pump beam carries no OAM ( = ) l 0 p , the OAM of the signal beam is transferred to the SFG field ( = ) l l SFG s . However, the spatial mode generated in the SFG process may be distorted with respect to the input mode as a result of spatial filtering due to: a) the limited spatial extent of the pump mode and b) the spatial bandwidth of the phase-matching function 51 . However, if the length of the nonlinear crystal is short compared to the diffraction length of the pump and signal beams, the spatial mode of the SFG yields a perfect frequency-converted replica of the signal field . A detailed analysis of the effects of spatial filtering is beyond the scope of this article, and we restrict our considerations to the conservation of the phase structure of superposition of modes with different values of OAM. For a detailed discussion of the spatial transfer function in SFG and SHG, see Refs 36, 52, 53 and 54.

Experimental Results
The experimental setup depicted in Fig. 1 consists of three stages. First, the input laser (803 nm) is manipulated in order to obtain the desired structured wave front. In the second stage, the frequency of the incident 803-nm beam is converted to a visible wavelength using SFG in a periodically poled Magnesium-doped lithium niobate (Mg:ppLN) crystal. Finally, the spatial characteristics of the sum-frequency mode are analyzed via far-field intensity measurements, and a coherent mode projection technique using a single-mode fiber. Each stage is discussed in more detail in the following.
Encoding the phase structure. The input mode was generated from a fiber-coupled 803-nm continuous-wave laser diode. The light emanating from the single-mode fiber was collimated its polarization state was set using a fiber polarization controller. The horizontally polarized 803-nm Gaussian beam was reflected off a spatial light modulator (SLM) and the first-order diffraction was modulated in phase and amplitude 55  to generate a desired spatial mode. Specifically, we generated either spiral phase = LG l p 0 modes 51 with a radial mode index p = 0, superpositions of LG l 0 and − LG l 0 , or HG nm modes, where n and m are the mode indices. A spatial filter removed the zero-th order diffraction from the SLM before directing the spatially modulated 803-nm beam to the frequency conversion stage. For each of the aforementioned input modes, we verified the expected far-field intensity profiles by inserting a CCD located in the Fourier plane of a 2-f imaging system (not depicted in Fig. 1).

Sum frequency generation.
The frequency converter is based on collinear type-0 SFG in a 10-mm-long Mg:ppLN crystal (Covesion Ltd.) with a poling period of 7.8 μ m. The interacting fields are all polarized vertically (parallel to the crystal's optical z-axis) and propagate along the x-axis in non-critical quasi-phase matching configuration. This avoids the introduction of wave-front distortions due to spatial walk-off, and it allows access to the largest nonlinear coefficient of LN (d 33 ), which leads to overall higher conversion efficiency. The crystal was maintained at a temperature of approximately 85°C, for SFG with quasi-phase matched center wavelengths of λ p = 1540 nm, λ s = 803 nm, and λ SFG = 527 nm.
A fiber-coupled 1540-nm continuous-wave laser diode was amplified to a power of approximately 60 mW via an Erbium-doped fiber amplifier (measured at the output of the single-mode fiber). A fiber polarization controller and a polarizing beam splitter cube were used to control the power of the pump beam. The pump beam was magnified and combined with the input beam via a dichroic mirror which was highly reflective at 803 nm and highly non-reflective at 1540 nm. The required vertical polarization states of the pump and input mode were set using zero-order half-wave plates (HWP). For the mixing process to occur in the Fourier plane, both beams were focused to the center of the nonlinear crystal using an achromatic lens with a focal length f = 100 mm. The beam waists of the Gaussian pump beam was ≈ w p 56 μ m and the waist parameter of the signal modes was ≈ µ w m 21 s , respectively. The SFG was collected using an f = 200-mm lens, and the pump light was guided to a beam dump using a dichroic mirror. The remaining pump light and the 803-nm signal were then blocked using a short-pass filter and a narrowband interference filter with a passband of approximately 10 nm, centered around 532 nm.

Detection.
A flip mirror directed the 527-nm sum-frequency beam to a module for either a phase-or an intensity measurement. The intensity measurements were performed in the far field using a CCD located at the Fourier plane of a 2-f imaging system. The phase structure of the outgoing beam was analyzed by projecting onto an appropriately tailored spatial mode [56][57][58] . The overlap with the LG 0 0 mode, which has a flat phase front, was evaluated by monitoring the power coupled into a single-mode fiber. For the projection onto a HG 10 mode, an additional phase mask was inserted before the single-mode fiber. The phase mask was a transparent microscope slide, which was inserted half-way into the beam path. The microscope slide was slightly tilted to obtain the required phase flip of π . The 803-nm input mode was characterized separately, using an additional phase mask and a single-mode fiber (not depicted in Fig. 1).
Frequency conversion of OAM superposition states. Our results can be visualized using a Poincaré sphere equivalent for OAM 59,60 (Fig. 2). The polar points on the sphere correspond to LG l 0 modes with =  l 1. The 803-nm light is blocked by an interference filter (IF) and the intensity profile of the sum-frequency field is analyzed using a CCD. The phase-structure of the SFG mode is analyzed using a phase mask and a single-mode fiber (SMF). The power coupled into the single-mode fiber is monitored using a power meter (PM). . These superpositions correspond to a HG 10 mode, which is rotated by an angle θ (π/ ) = ( ) [HG 2 HG 0 ] 10 01 . By changing the relative phase and magnitude of the OAM components one can access states distributed over the entire Poincaré sphere. For all phase profiles applied to the input beam, the 527-nm frequency-converted SFG field exhibits the same characteristic far-field intensity profile as the 803-nm input mode (inset Fig. 2). Similar results were obtained for measurements in the image plane (not depicted in Fig. 2.). The fact that the angular orientation of the sum-frequency mode is the same as that of the signal beam demonstrates that the relative phase of the OAM states is conserved in the frequency-conversion process.
For 60 mW of pump power (47 mW impinging on the nonlinear crystal), we observed power-conversion efficiencies (η p = P SHG /P signal ) of 0.11% for the LG 0 0 input mode (not depicted in Fig. 2), 0.09% for the ± LG 0 1 modes, and 0.08% for the HG 01 and HG 10 modes. The quantum conversion efficiencies η η = ω ω p S SFG were 0.07% for the LG 0 0 input mode, 0.06% for the ± LG 0 1 modes, and 0.05% for HG 01 and HG 10 modes, respectively. These results were obtained for a moderate pump power, and the setup can be expected to achieve quantum efficiencies > 1% for a pump power of  1 W. At 1550 nm, pump powers >1 W are readily obtained using commercially-available Erbium-doped fiber amplifiers. Achieving higher conversion efficiencies is the subject of further research, and we believe that significant The fidelity of the frequency-converted spatial intensity patterns demonstrates the feasibility of coherently frequency converting complex coherent spatial patterns, as well as higher-dimensional quantum systems.
improvements can be expected by placing the nonlinear crystal inside an optical cavity as was done by Zhou et al. 49 , or by optimizing the alignment and confocal parameters of the pump-and signal beams.
In this proof-of-concept experiment, however, our primary goal was to demonstrate that phase coherence was maintained in the frequency-conversion process. While this claim is already supported by the results discussed in the preceding paragraph, we also performed phase-dependent measurements using a coherent mode projection technique. As a first step, we evaluated the overlap of both the input beam and the SFG beam with the approximately Gaussian mode of a single-mode fiber. Figure 3(a) depicts the normalized fiber-coupled power for various phase profiles applied to the input beam via the SLM. As expected, only the flat phase front of the LG 0 0 input beam led to efficient coupling into the respective single-mode fiber. Conversely, , HG HG 01 10 , and ± LG 0 1 modes showed negligible overlap with the LG 0 0 projection mode. Similar behavior was observed when a phase mask with a π phase step was added before the projection fiber. This approximately corresponds to the projection onto the anti-symmetric HG 10 mode (Fig. 3(b)). With the phase mask in place, the LG 0 0 mode resulted in a negligible amount of power coupling into the single mode fiber. In this case, maximal single-mode fiber-coupling efficiency was observed for a HG 10 mode, for which the phase mask effectively flattens the wave front. The HG 01 mode, on the other hand, showed negligible overlap with the projection mode. This behavior is expected due to the orthogonality of the HG modes. Since the ± LG 0 1 modes are equal superpositions of HG 10 and HG 01 modes we expect the single-mode fiber-coupling to decrease by 50% with respect to the HG 10 mode. The experimental fiber-coupling efficiencies were slightly unbalanced for LG 0 1 and − LG 0 1 , due to misalignment of the SLM and the projection fiber (see Fig. 3(b)). In summary, the same characteristic behavior was observed for both the input and SFG modes. From these results we can conclude that the phase relationship between different OAM modes is conserved, which demonstrates the suitability of the scheme for coherently converting OAM superposition states for applications in quantum information processing.
Frequency conversion of higher-order spatial modes. In order to extend these results to higher OAM values and further assess the feasibility of applying the technique, for example, in the coherent imaging of more complex phase structures, we also evaluated the frequency-converted intensity profiles for several higher-order spatial modes. The spatial intensity profiles (Fig. 4) of the 527-nm mode exhibited a high degree of similarity with the 803-nm input mode. However, for higher mode numbers aberrations became apparent. This is the result of crystal inhomogeneity, beam clipping effects at the limited aperture of the nonlinear crystal, as well as spatial filtering effects due to the beam waist of the pump laser. This effect could, however, be mitigated using both a larger crystal aperture, as well as a larger beam waist. Determining the pump focus parameters which lead to the highest conversion-efficiency for a particular higher-order spatial mode, as well as the transfer function for more general distributions of transverse momenta should be the subject of further research.

Discussion
We have demonstrated coherent frequency conversion of structured light, specifically Laguerre-Gauss and Hermite-Gauss modes, from the near infrared to the visible via the process of sum-frequency mixing in a periodically poled Mg:LN crystal. Our results demonstrate conclusively that the relative phase between Laguerre-Gauss modes with different OAM content is maintained in the frequency-conversion process. This establishes the feasibility of frequency converting quantum information encoded in OAM superposition states for applications in higher-dimensional quantum information processing protocols.
More generally, this can be seen as a method for trans-frequency imaging of a phase-only object, and we believe that more complex phase-structured optical beams may be converted using similar techniques. This could be of great relevance to novel applications in microscopy, since it allows acquiring phase information about an object from light at a wavelength different from that used to illuminate the object 61 .
However, further improvements are necessary in order to obtain higher conversion efficiencies as required for wide-scale practical applicability in different scenarios. Nevertheless, our results demonstrate the experimental feasibility of proof-of-concept experiments involving the coherent frequency conversion of multi-dimensional quantum information encoded in the spatial degree of freedom of (entangled) single photons, which could be of great significance for future applications in quantum information processing and sensing.
The approach used in this article could also be applied to optical parametric amplification, thus enabling the generation of desired spatial modes with high power from a weak seed field. Furthermore, with rapid advancements in nonlinear materials science, it is also foreseeable that similar techniques may be used to mediate parametric photon-photon interactions for complete Bell-state analysis in high-dimensional spatial teleportation protocols and entanglement swapping 62,63 .