## Abstract

The geometric phase is a common phenomenon in a variety of physical systems, with significant applications in optics. We report the first experimental demonstration of the adiabatic geometric phase in nonlinear frequency conversion, wherein the coupling between the signal and idler frequencies constitutes the intrinsic two-level dynamics of the system. We observe a variety of effects associated with the geometric phase, including the adiabatic broadening of bandwidth, asymmetric transmission for opposite propagation directions, conjugation of the phase for orthogonal eigenstates, and the nonreciprocity associated with the pump field bias. Our work paves the way towards all-optically controlled geometric phase elements for wavefront shaping, isolation, guiding, and quantum optical applications, harnessing spectral and spatial correlations.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

When a two-level system performs a closed trajectory in its parameter space, the final state accumulates a phase factor dependent on the trajectory, in addition to its dynamical phase attributed to propagation [1,2]. This geometric phase was first discovered by Pancharatnam for polarized light [3] and later by Berry for quantum mechanical systems [4]. This phenomenon has since had a profound effect on physics. Its applications range from fundamental quantum mechanics [5], condensed matter physics [6], and, of course, in the field of light optics. For circularly polarized light, the so-called Pancharatnam–Berry (PB) phase, accumulated when the photon helicity is changed from positive to negative using a transversely varying half-wave plate element, is widely used for beam shaping [7–9], beam shearing [10], guiding [11], holography [12], and for encoding both classical [13] and quantum [14] information.

The PB phase has been utilized in the field of nonlinear optics using nonlinear photonic metasurfaces [15–18] for controlling the phase of the generated higher harmonics. In these schemes, the polarization of the nonlinear response is continuously controlled by varying the orientation angle $\theta $ of the nanoantennas hosting a strong local nonlinearity. While the fundamental frequency acquires the PB phase of $\pm 2\theta $, the generated $n$th harmonic attributed to the locally induced nonlinear dipole moment can acquire a geometric phase of $\pm (n\pm 1)\theta $, depending on the original pump helicity as well as on the helicity of the generated light [18]. These methods are versatile, allowing for arbitrary two-dimensional holograms in the generated harmonic using thin elements, as they essentially rely on the polarization degree of freedom. However, they suffer from very low conversion efficiencies due to the small interaction volumes, albeit the nonlinear susceptibility itself is typically enhanced.

Nonlinear photonic crystals (NLPCs), on the other hand, can offer a significantly higher and broadband conversion efficiency, for example, by the use of adiabatic processes [19]. However, the control over the polarization of the susceptibility tensor in NLPCs is limited to the binary poling methods used, for example, in quasi-phase-matching (QPM) techniques [20]. Unfortunately, this eliminates the possibility for a nonlinear PB phase scheme in NLPCs. A different approach, incorporating both high efficiency, bandwidth, and versatility, without relying on the polarization degree of freedom, should therefore be pursued.

In this paper, we report the first experimental study of the adiabatic geometric phase in nonlinear frequency conversion, where the *spectral* degree of freedom is used instead of the light’s polarization. We employ the second-order process of sum-frequency generation (SFG) and, following previous works [21–23], use electric field poling to engineer the interaction. Our experimental results test the phenomena predicted for the geometric phase in this system, such as broadened acceptance bandwidth, asymmetric transmission, and nonreciprocity.

## 2. THEORETICAL MODEL

The analogy between SFG and two-level systems has been investigated over the past few years [19,24,25]. Under the slowly varying envelope and undepleted pump approximations, the two bare eigenstates of the system are the signal and idler frequencies, ${\omega}_{i}$ and ${\omega}_{s}$, respectively, separated in energy by the pump frequency, ${\omega}_{p}={\omega}_{s}-{\omega}_{i}$. The analog of the detuning is proportional to the phase mismatch $\mathrm{\Delta}k={k}_{i}+{k}_{p}-{k}_{s}-2\pi /\mathrm{\Lambda}$, where ${k}_{j},j=i,s,p$ are the idler, signal, and pump wavenumbers, respectively, and $\mathrm{\Lambda}$ is the QPM period [20]. In addition, the coupling parameter is $\kappa \propto {\chi}^{(2)}\sqrt{{I}_{p}}$, where ${\chi}^{(2)}$ and ${I}_{p}$ are the second-order susceptibility coefficient and pump intensity profile, respectively. Finally, the system dynamics is described with respect to the propagation $z$ coordinate.

The adiabatic geometric phase in SFG has been rigorously derived in Ref. [24], and here we follow the main results. In the framework of spin-1/2 systems (where the idler and signal frequencies are equivalent to the two opposite spin states in the $z$ direction), one can define an analog magnetic field, $\mathit{B}$, after a coordinate transformation to a rotating frame,

A two-level system subject to an adiabatically changing Hamiltonian preserves its instantaneous eigenstates [5]. In the context of nonlinear optics, this trait has been utilized for broadband and robust frequency conversion [19,27]. In addition, if the Hamiltonian is rotated adiabatically in parameter space, an additional phase is gained, depending only on the geometric trajectory of the Hamiltonian. If the latter is a closed trajectory, then the value of the geometric phase, $\gamma $, is given in terms of the solid angle, $\mathrm{\Omega}$, circumvented by the path,

where the $\mp $ signs correspond to the two different eigenstates, or, independently, to the direction of the trajectory, clockwise or counterclockwise. This implies that the geometric phase is manifested in a conjugate manner for*orthogonal eigenstates*and

*opposite propagation directions*.

## 3. EXPERIMENTAL RESULTS

In order to experimentally demonstrate these concepts in SFG, we designed three 10 mm long Mg:CLN nonlinear crystals. In all designs, the poling parameters $D$, $\mathrm{\Lambda}$, and $\phi $ are varied along the propagation direction in a specific manner [24] to induce an adiabatic trajectory of the analogous magnetic field [Eq. (1)] and a corresponding geometric phase. We considered two types of trajectories–a wedge that connects the north and south poles via two different open paths [Fig. 1(a)]; and a circular trajectory around a fixed vector in parameter space [Fig. 3(b)]. For the pump beam, we used a 1064.5 nm Nd:YAG pulsed laser with a 10 kHz repetition rate, a pulse length of 4.5 nsec and 1 W average power. Additionally, we used two CW lasers: an He:Ne laser at 632.8 nm for the signal frequency with a power of 8 mW, and an infrared tunable laser source (TLS) centered at 1550 nm for the idler frequency, amplified to have a power of 22 mW. The crystals were kept at a constant temperature of 54°C.

In the first design, we fabricated two adjacent parallel paths on the same crystal, each employing an adiabatic trajectory from the north pole of the parameter space surface (corresponding to the idler frequency on the Bloch sphere) to the south pole (signal frequency). This transition was achieved by varying the poling period from 11.15 to 12.49 μm, and simultaneously varying the duty cycle from 30% (approximately 3.5 μm for the poled region width, a value limited by the fabrication constraints) to 50% (see Supplement 1 for more information). The two paths differ from one another by an angle $\mathrm{\Delta}\phi $, which defines a closed wedge on the surface in parameter space [Fig. 1(a)]. We choose $\mathrm{\Delta}\phi =\gamma =\pi $ to gain a $\pi $ phase difference between the two paths, which in the far field approximates a Hermite–Gauss 01 (HG01) beam.

The parameter-space surface should not be confused with the Bloch sphere, which is the mapping of the Hilbert space to the surface of a sphere. Parameter space, however, may accommodate a more general surface, since the size of $\mathit{B}$ does not have to be kept constant. For instance, by varying the poling period (the $z$ component of $\mathit{B}$) more dominantly near the crystal edges, the parameter space surface becomes elongated while staying azimuthally symmetric [Fig. 1(a)]. This is typical of the adiabatic process in nonlinear optics [19]. Changing the input wavelength, and hence the phase mismatch $\mathrm{\Delta}k$, can be geometrically understood as shifting the parameter space surface up or down with respect to the origin. Hence, if the origin is enclosed by an elongated surface as in Fig. 1(a), a larger input frequency tolerance is allowed [as compared to the usual spherical shape in ordinary SFG, or in the circular rotation scheme, Fig. 3(b)], resulting in a broader conversion bandwidth. As the surface becomes more elongated, corresponding to a larger variation of the phase mismatch $\mathrm{\Delta}k$, the spectral bandwidth becomes larger. Additionally, the photon conversion efficiency can approach unity if strong pump intensities are used. The implementation of this method requires satisfying the adiabatic conditions, which depend on the length of the crystal, the pump intensity, and the rate of change of the poling period. These and other considerations have been discussed in detail in Ref. [19].

The experimental setup is illustrated in Fig. 1(b). We sum the idler frequency at 1550 nm and the pump wave at 1064.5 nm, and measure the far-field intensity pattern at 631 nm (after filtering) using a CCD camera. We observe the resulting HG01-like mode, as can be seen in Fig. 1(c). The measured signal power was 30.44 mW. This corresponds to a maximal photon number conversion efficiency ($\eta ={N}_{s}/{N}_{i}$, where ${N}_{j},j=i,s$ is the photon number of the idler and signal fields, respectively) at 1551 nm of $\eta =64\%$, whereas the simulated efficiency (calculated using the split-step Fourier method [28]) was $\eta =65\%$. The FWHM of the measured efficiency distribution was $\sim 8\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$, which is 4 times larger than a periodically poled crystal of the same length [see Fig. 1(d)].

The second design implements a nonlinear cylindrical lens using the wedge scheme as in the previous design [Fig. 1(a)], converting from idler to signal (and vice versa). The crystal induces a quadratic geometric phase that varies along the transverse $x$ dimension, e.g., $\gamma (x)=\mathrm{\Delta}\phi (x)=a{x}^{2}$, where $a={10}^{3}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{rad}/{\mathrm{mm}}^{2}$. The poling period and duty cycle were varied along the crystal in the same manner as in the first design. The resulting focal length of the lens is $f=\pi n/\lambda a$, where $n$ and $\lambda $ are the refractive index and the wavelength of the converted beam, respectively. We measured the beam propagation from the crystal facets, for four cases, corresponding to all combinations of the crystal orientation relative to the pump direction and to the input wavelength (632.8 and 1550 nm); see Fig. 2.

As a result of Eq. (3), we expect to observe two typical traits of the geometric phase. The first is asymmetric transmission, reported recently for nonlinear PB metasurfaces (wherein the light polarization was used for generating the geometric phase, as opposed to the scheme discussed here [29]). This means that the lens has an opposite effect on counterpropagating beams: by changing the crystal orientation relative to the pump direction, the converted beam will be either converging or diverging. This happens because the trajectory in parameter space is determined by the crystal orientation, changing from clockwise to counterclockwise, thus flipping the sign of the phase.

The effect can be seen by comparing Figs. 2(c) and 2(d) or Figs. 2(e) and 2(f). The beam propagation was imaged after the exit facet of the crystal, located at $z=10\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$ (c)–(d) or after the entrance facet $z=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$ (e)–(f).

The second trait is the conjugation of the accumulated phase when starting at orthogonal eigenstates. By changing the input wavelength, the converted beam changes from converging to diverging. The effect can be seen by comparing Figs. 2(c) and 2(e) or Figs. 2(d) and 2(f). Notice that the focal length of the signal and idler varies by a factor of ${\lambda}_{i}/{\lambda}_{s}\cong 2.5$, and therefore the focal point appears inside the crystal for the case in Fig. 2(b).

In the final design, we employ a different type of trajectory in parameter space, wherein both the initial and the final points belong to the same eigenstate [24]. As can be seen in Fig. 3(b), in this trajectory, the magnetic field analog, $\mathit{B}$, precesses on a circle about a unit vector at angle $\mathrm{\Theta}$ relative to the $z$ axis. The circle encloses a solid angle of $\mathrm{\Omega}=\pi (1-\mathrm{cos}\text{\hspace{0.17em}}\mathrm{\Theta})$ on the surface, which coincides with the Bloch sphere. In this case, there is no net frequency conversion; however, the output eigenstate accumulates a geometric phase given by Eq. (3). In our design, we fabricated two adjacent paths having a $\pi $ geometric phase difference: in the first path $\mathit{B}$ precesses clockwise, gaining a $\pi /2$ phase ($\mathrm{\Theta}=\pi /3$), while in the second path, the precession is counterclockwise on the same trajectory, giving a geometric phase of $-\pi /2$. The poling period is varied between 11.84 and 11.75 μm and back; the duty cycle is varied between 30% to 50% and back, as described in Fig. 2 in the Supplement 1. The resulting output beam, therefore, is expected to behave similarly to an HG01 beam, as in the first design, but in this case the shaping is obtained for the original *input* wavelength.

This process also demonstrates an intriguing property of the nonlinear geometric phase, namely, the breaking of time reversal symmetry. As in other optical systems based on the Faraday effect [30], the Fizeau effect [31] or a traveling-wave temporal modulation of the refractive index [32], having an external *bias* odd under time reversal, the interaction is nonreciprocal [33]. In the case of SFG, the bias is the pump field’s momentum, which is unidirectional. Reversing the direction of the input idler or signal fields relative to the pump results in a highly nonphase-matched interaction ($\mathrm{\Delta}k$ comparable to optical wave vectors), for which the conversion efficiency ($|\eta |\sim 1/\mathrm{\Delta}kL$, $L$ the crystal length) effectively vanishes, allowing for high isolation after spectral filtering. This idea was first theoretically proposed in [34] and then later in [35] for frequency-conversion-based schemes. In our experiment, we expect the nonlinear crystal to act as a frequency-retaining, nonreciprocal spatial mode converter: entering from one side, the beam changes its spatial mode from HG00 to HG01, keeping its original frequency. Entering from the other side relative to the pump, no observable interaction is expected owing to the large phase mismatch, and the beam retains both its frequency and spatial mode.

In order to measure the spatial conversion of the input CW frequency using a pulsed pump field, it was necessary to temporally correlate the input field with the pump pulse. For this manner, we used an additional adiabatic KTP crystal designed for the SFG conversion of 1550–631 nm prior to the geometric phase crystal. As can be seen in Fig. 3(c), we adiabatically convert the CW idler (1550 nm) to a pulsed signal (631 nm), correlated with the pulsed pump (1064.5 nm), after which the original CW idler is filtered. The two pulsed fields then enter the geometric phase crystal, wherein the pulsed signal is adiabatically rotated back to itself as described above, and the resulting spatial mode is imaged onto a CCD camera after the exit facet of the crystal. In Fig. 3(d), we compare two cases: in the first, the pump field is filtered out before entering the geometric phase crystal, and the signal retains its original Gaussian mode; in the second, both pump and signal fields are copropagating in the crystal, and the signal’s spatial mode is hence converted. In this manner, we indirectly observe the expected nonreciprocity of the system, since the outcome of the former case is identical to the counterpropagation of the signal pulse relative to the pump, wherein the nonlinear interaction is diminished well below the noise level of the detection system. The poling has a small but visible effect on the refractive index, as can be seen in Fig. 3(d1), where the gap between the two adjacent paths [e.g., as in Fig. 1(b)] can be seen in the beam cross section.

## 4. CONCLUSION

In conclusion, we have experimentally demonstrated for the first time the adiabatic geometric phase in nonlinear frequency conversion and observed its properties. As a possible extension of this work, the concept of beam shaping in nonlinear optics can be generalized to fully control the transverse plane of the shaped beam using the adiabatic rotation schemes, wherein the recent technological advancements in three-dimensional NLPCs [36,37] offer promising outlooks. One such exciting possibility is the experimental realization of adiabatic geometric phase masks for generating vortex beams that carry orbital angular momentum [24]. Another possible outlook is to extend our formalism to a three-level system, using a cascaded SFG process in quasi-periodic NLPCs [38], with more than one pump field. For example, in a ladder configuration with ${\omega}_{1}<{\omega}_{2}<{\omega}_{3}$, one has ${\omega}_{2}={\omega}_{1}+{\omega}_{p1}$ and ${\omega}_{3}={\omega}_{2}+{\omega}_{p2}$, where ${\omega}_{p1},{\omega}_{p2}$ denote the two pump frequencies, and where the quasi-periodic crystal provides the phase-matching conditions for the multiple processes [39]. Under certain constraints, such systems were shown to exhibit spin-1 dynamics [40], and by varying the couplings along the crystal, in a similar manner to what we described above for the two-level SFG process, it should be possible to accumulate a geometric phase for each of the corresponding eigenstates, having the three spin-1 eigenvalues $m=-1,0,1$, as $\gamma =-m\mathrm{\Omega}$ (thus generalizing Eq. (2) for spin-1). Interestingly, unlike the spin-1 photon helicity [41,42] taking only the two values of $m=\pm 1$ due to the photon being a massless particle, this proposed system can also occupy the $m=0$ state. The SFG geometric phase can have an important role in future applications, including geometric phase-based guiding of both the signal and idler using a geometric phase lens array [11], as well as in quantum-optics applications, wherein the frequency degree of freedom can be entangled with spatial modes such as orbital angular momentum [14] or the photon’s path [40].

## Funding

Israel Science Foundation (1415/17).

## Acknowledgment

We acknowledge HC-Photonics for the fabrication of the crystal designs. A. K. acknowledges support by the Adams Fellowship Program of the Israel Academy of Science and Humanities. S. T.-M. acknowledges support by the Ministry of Science Shulamit Aloni scholarship.

Please see Supplement 1 for supporting content.

## REFERENCES AND NOTES

**1. **E. Cohen, H. Larocque, F. Bouchard, F. Nejadsattari, Y. Gefen, and E. Karimi, “Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond,” Nat. Rev. Phys. **1**, 437 (2019). [CrossRef]

**2. **M. V. Berry, “Geometric phases,” Word Press, 2018, https://michaelberryphysics.files.wordpress.com/2018/03/berryd.pdf.

**3. **S. Pancharatnam, “Generalized theory of interference, and its applications,” Proc. Indian Acad. Sci. A. **44**, 247–262 (1956). [CrossRef]

**4. **M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London Ser. A **392**, 45–57 (1984). [CrossRef]

**5. **J. J. Sakurai and J. Napolitano, *Modern Quantum Mechanics* (Addison-Wesley, 2011).

**6. **Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature **438**, 201–204 (2005). [CrossRef]

**7. **Z. Bomzon, V. Kleiner, and E. Hasman, “Pancharatnam–Berry phase in space-variant polarization-state manipulations with subwavelength gratings,” Opt. Lett. **26**, 1424–1426 (2001). [CrossRef]

**8. **L. Marrucci, C. Manzo, and D. Paparo, “Pancharatnam–Berry phase optical elements for wave front shaping in the visible domain: switchable helical mode generation,” Appl. Phys. Lett. **88**, 221102 (2006). [CrossRef]

**9. **R. C. Devlin, A. Ambrosio, D. Wintz, S. L. Oscurato, A. Y. Zhu, M. Khorasaninejad, J. Oh, P. Maddalena, and F. Capasso, “Spin-to-orbital angular momentum conversion in dielectric metasurfaces,” Opt. Express **25**, 377–393 (2017). [CrossRef]

**10. **L. A. Alemán-Castaneda, B. Piccirillo, E. Santamato, L. Marrucci, and M. A. Alonso, “Shearing interferometry via geometric phase,” Optica **6**, 396–399 (2019). [CrossRef]

**11. **S. Slussarenko, A. Alberucci, C. P. Jisha, B. Piccirillo, E. Santamato, G. Assanto, and L. Marrucci, “Guiding light via geometric phases,” Nat. Photonics **10**, 571–575 (2016). [CrossRef]

**12. **T. Zhan, J. Xiong, Y.-H. Lee, R. Chen, and S.-T. Wu, “Fabrication of Pancharatnam–Berry phase optical elements with highly stable polarization holography,” Opt. Express **27**, 2632–2642 (2019). [CrossRef]

**13. **G. Milione, T. A. Nguyen, J. Leach, D. A. Nolan, and R. R. Alfano, “Using the nonseparability of vector beams to encode information for optical communication,” Opt. Lett. **40**, 4887–4890 (2015). [CrossRef]

**14. **T. Stav, A. Faerman, E. Maguid, D. Oren, V. Kleiner, E. Hasman, and M. Segev, “Quantum entanglement of the spin and angular momentum of photons using metamaterials,” Science **361**, 1101–1104 (2018). [CrossRef]

**15. **G. Li, S. Zhang, and T. Zentgraf, “Nonlinear photonic metasurfaces,” Nat. Rev. Mater. **2**, 17010 (2017). [CrossRef]

**16. **N. Segal, S. Keren-Zur, N. Hendler, and T. Ellenbogen, “Controlling light with metamaterial-based nonlinear photonic crystals,” Nat. Photonics **9**, 180–184 (2015). [CrossRef]

**17. **M. Tymchenko, J. S. Gomez-Diaz, J. Lee, N. Nookala, M. A. Belkin, and A. Alù, “Gradient nonlinear Pancharatnam–Berry metasurfaces,” Phys. Rev. Lett. **115**, 207403 (2015). [CrossRef]

**18. **G. Li, S. Chen, N. Pholchai, B. Reineke, P. W. H. Wong, E. Y. B. Pun, K. Wai Cheah, T. Zentgraf, and S. Zhang, “Continuous control of the nonlinearity phase for harmonic generations,” Nat. Mater. **14**, 607–612(2015). [CrossRef]

**19. **H. Suchowski, G. Porat, and A. Arie, “Adiabatic processes in frequency conversion,” Laser Photon. Rev. **8**, 333 (2014). [CrossRef]

**20. **R. W. Boyd, *Nonlinear Optics* (Academic, 2008).

**21. **A. Shapira, L. Naor, and A. Arie, “Nonlinear optical holograms for spatial and spectral shaping of light waves, Sci. Bull. **60**, 1403–1415 (2015). [CrossRef]

**22. **S. Trajtenberg-Mills and A. Arie, “Shaping light beams in nonlinear processes using structured light and patterned crystals,” Opt. Mater. Express **7**, 2928–2942 (2017). [CrossRef]

**23. **S. Trajtenberg-Mills, I. Juwiler, and A. Arie, “On-axis shaping of second-harmonic beams,” Laser Photon. Rev. **9**, L40–L44 (2015). [CrossRef]

**24. **A. Karnieli and A. Arie, “Fully controllable adiabatic geometric phase in nonlinear optics,” Opt. Express **26**, 4920–4932 (2018). [CrossRef]

**25. **A. Karnieli and A. Arie, “All-optical Stern–Gerlach effect,” Phys. Rev. Lett. **120**, 053901 (2018). [CrossRef]

**26. **For first order QPM, the coupling is proportional to $\text{sin}\hspace{0.17em}\pi D$, where $D$ is the duty cycle.

**27. **P. Krogen, H. Suchowski, H. Liang, N. Flemens, K.-H. Hong, F. X. Kärtner, and J. Moses, “Generation and multi-octave shaping of mid-infrared intense single-cycle pulses,” Nat. Photonics **11**, 222–226(2017). [CrossRef]

**28. **O. V. Sinkin, R. Holzlöhner, J. Zweck, and C. R. Menyuk, “Optimization of the split-step Fourier method in modeling optical-fiber communications systems,” J. Lightwave Technol. **21**, 61–68 (2003). [CrossRef]

**29. **N. Shitrit, J. Kim, D. S. Barth, H. Ramezani, Y. Wang, and X. Zhang, “Asymmetric free-space light transport at nonlinear metasurfaces,” Phys. Rev. Lett. **121**, 046101 (2018). [CrossRef]

**30. **B. E. A. Saleh and M. C. Teich, *Fundamentals of Photonics* (Wiley-Interscience, 2007).

**31. **A. Lipson, S. G. Lipson, and H. Lipson, *Optical Physics* (Cambridge University, 2011).

**32. **H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. **109**, 033901 (2012). [CrossRef]

**33. **C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. **10**, 047001 (2018). [CrossRef]

**34. **K. Wang, Y. Shi, A. S. Solntsev, S. Fan, A. A. Sukhorukov, and D. N. Neshev, “Non-reciprocal geometric phase in nonlinear frequency conversion,” Opt. Lett. **42**, 1990–1993 (2017). [CrossRef]

**35. **T. Li, K. Abdelsalam, S. Fathpour, and J. B. Khurgin, “Wide bandwidth, nonmagnetic linear optical isolators based on frequency conversion,” in *Conference on Lasers and Electro-Optics* (OSA, 2019), paper FW3B.7.

**36. **T. Xu, K. Switkowski, X. Chen, S. Liu, K. Koynov, H. Yu, H. Zhang, J. Wang, Y. Sheng, and W. Krolikowski, “Three-dimensional nonlinear photonic crystal in ferroelectric barium calcium titanate,” Nat. Photonics **12**, 591–595 (2018). [CrossRef]

**37. **D. Wei, C. Wang, H. Wang, X. Hu, D. Wei, X. Fang, Y. Zhang, D. Wu, Y. Hu, J. Li, S. Zhu, and M. Xiao, “Experimental demonstration of a three-dimensional lithium niobate nonlinear photonic crystal,” Nat. Photonics **12**, 596–600 (2018). [CrossRef]

**38. **A. Arie and N. Voloch, “Periodic, quasi-periodic, and random quadratic nonlinear photonic crystals,” Laser Photon. Rev. **4**, 355–373 (2010). [CrossRef]

**39. **R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. **95**, 133901 (2005). [CrossRef]

**40. **A. Karnieli and A. Arie, “Frequency domain Stern–Gerlach effect for photonic qubits and qutrits,” Optica **5**, 1297–1303 (2018). [CrossRef]

**41. **R. Y. Chiao and Y.-S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. **57**, 933 (1986). [CrossRef]

**42. **S. Weinberg, *The Quantum Theory of Fields* (Cambridge University, 1995).