### Mode Sorters

Mode Sorters are devices that can decompose an arbitrary optical field into a complete set of orthogonal modes. A specific example of a mode sorter is a device that can decompose a light field into the members of the Laguerre-Gauss modes. Mode sorting is an enabling technology for many applications in photonics, such as optical communication, which has been our primary interest. For technical reasons, it is more difficult to construct mode sorters for some degrees of freedom than for others; for example, it is more difficult to decompose into the radial Laguerre-Gauss modes than into its azimuthal modes. In earlier work, we showed how to construct a radial mode sorter [1]. In the past year we extended this work by constructing a mode sorter for all transverse degrees of freedom of a light field and using this device in a quantum communication system [2].

- Sorting Photons by Radial Quantum Number, Y. Zhou, M. Mirhosseini, D. Fu, J. Zhao, S. M. H. M. Rafsanjani, A. E. Willner, and R. W. Boyd, Phys. Rev. Letters 119, 263602 (2017).
- Using all transverse degrees of freedom in quantum communications based on a generic modesorter, Y. Zhou, M. Mirhosseini, S. Oliver, J. Zhao, S. M. H. Rafsanjani, M. P. J. Lavery, A. E. Willner, and R. W. Boyd, Optics Express 27, 10383-10394 (2019). (Designated as an Editor’s Pick.)

### Optical Metamaterials

Optical metamaterials are materials that possess optical properties dissimilar to those of any naturally occurring material. Earlier, we fabricated a nonlinear metasurfaces consisting of gold nanoantennas sitting on a substrate of indium tin oxide (ITO), and we showed that such a material displays a nonlinear optical response many orders of magnitude larger than those of naturally occurring materials. We have also designed a metasurface consisting of mutually interacting nanoantennas that produce a sequence of high-Q resonances. We have shown numerically that such a structure can generate a strong second-harmonic output field.

- Large optical nonlinearity of nanoantennas coupled to an epsilon-near-zero material, M. Zahirul Alam, S. A. Schulz, J. Upham, I. De Leon and R. W. Boyd, Nature Photonics 12, 79–83 (2018).
- Efficient nonlinear metasurfaces by using multiresonant high-Q plasmonic arrays, M. J. Huttunen, O. Reshef, T. Stolt, K. Dolgaleva, R. W. Boyd, and M. Kauranen, Journal of the Optical Society of America B 36, E30-E35 (2019).

### Quantum Imaging

Quantum Imaging entails the use of quantum procedures to enhance the quality of the image formed by an optical system. In the past year we developed means to achieve the quantum-optimal longitudinal resolution of an idealized image consisting of two mutually incoherent point sources [1]. We have also developed a new means of image formation that combines aspects of ghost imaging and interaction-free measurements that can perform at near-vanishing levels of illumination [2].

- Quantum-limited estimation of the axial separation of two incoherent point sources, Y. Zhou, J. Yang, J. D. Hassett, S. M. H. Rafsanjani, M. Mirhosseini, A. N. Vamivakas, A. N. Jordan, Z. Shi, and R. W. Boyd, Optica 6, 534-541 (2019).
- Interaction-Free Ghost-Imaging of Structured Objects, Y. Zhang, A. Sit, F. Bouchard, H. Larocque, F. Grenapin, E. Cohen, A. C. Elitzur, J. L. Harden, R. W. Boyd, and E. Karimi, Optics Express 27, 2212-2224 (2019).).

### Quantum Radiometry

We have developed a means based on the properties of the nonlinear optical process of spontaneous parametric downconversion to provide an absolute calibration of the spectral irradiance of the output of a spectrophotometer [1].

1. A primary radiation standard based on quantum nonlinear optics, S. Lemieux, E. Giese, R. Fickler, M. V. Chekhova, and R. W. Boyd, Nature Physics 15, 529 (2019).

### Secure Quantum Communication

The quantum properties of light can be used to develop protocols for communication that are entirely immune to eavesdropper attacks. Quantum communication is perhaps the most advanced of the quantum technologies. Our work in this area has extended from engineering studies to fundamental physics studies. Our work in the past year has involved constructing and characterizing an underwater quantum communication channel [1], which could be of great use in national defense, for example, in communication between a surface ship and a submarine. We also studied the influence of state-dependent diffraction [2] on the integrity of a transverse-mode-multiplexed quantum communication system. This concern is of special importance for systems that employ a large state space, for which case it is likely that not all states would have the same far-field intensity distribution. We show that this problem can be minimized by pre-compensating each state so that they have the same loss due to diffraction.

- Characterization of an underwater channel for quantum communications in the Ottawa River, F. Hufnagel, A. Sit, F. Grenapin, F. Bouchard, K. Heshami, D. England, Y. Zhang, B. J. Sussman, R. W. Boyd, G. Leuchs, and E. Karimi, Optics Express 27,26346-26354 (2019).
- Performance analysis of d-dimensional quantum cryptography under state-dependent diffraction, J. Zhao, M. Mirhosseini, B. Braverman, Y. Zhou, S. M. H. Rafsanjani, Y. Ren, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and R. W. Boyd, Phys. Rev. A 100, 032319 (2019).

### Nonlinear Optics

We have conducted studies of basic nonlinear optical processes. Traditionally, nonlinear optical processes have been studied using focused laser beams. The illumination of the interaction region usually comes from a solid angle W much smaller than 2p steradians, the solid angle subtended by a hemisphere. Most theoretical treatments of nonlinear optics consequently treat the nonlinear interaction through use of the paraxial approximation. In our study, we have for the first time studied the opposite limiting case, where the sample is excited coherently from all directions by an incoming spherical wave that subtends a solid angle of 4p steradians. Nonlinear optical processes behave very differently in this limit. There is no phase-matching requirement, because the nonlinear signal comes primarily from the focal region, which in this case has a size of approximately the wavelength of light. We find that nonlinear processes consequently become very efficient in this limit [1].

We have studied the process of adiabatic wavelength conversion [2] in a highly nonlinear material, indium tin oxide excited at a wavelength where the real part of its dielectric permittivity vanishes, the so-called epsilon-near-zero (ENZ) condition. We find that the wavelength range over which the output wave can be tuned is much larger for ENZ regions than had previously been studied under non-ENZ conditions. We have also performed a theoretical study of the nonlinear propagation of THz pulses [3]. THz propagation shows effect qualitatively different from the propagation of visible light because the wavelength of THz waves is so large the diffraction effects are dominant and the paraxial is not valid. Moreover, THz nonlinearities tend to be very much stronger than nonlinearities at optical frequencies. We have also studied the nonlinear propagation of few-cycle optical pulses [4]. A key finding is that self-focusing effects tend to be strongly suppressed in this circumstance, as a result of pulse broadening through the process of group velocity dispersion.

- Nonlinear optics with full three-dimensional illumination, R. Penjweini, M. Weber, M. Sondermann, R. W. Boyd, and G. Leuchs, Optica 6, 878-883 (2019).
- Broadband frequency translation through time refraction in an epsilon-near-zero material, Y. Zhou, M. Z. Alam, M. Karimi, J. Upham, O. Reshef, C. Liu, A. E. Willner and R. W. Boyd, Nature Communications 11, 2180 (2020).
- Propagation of broadband THz pulses: effects of dispersion, diffraction and time-varying nonlinear refraction, P. Rasekh, M. Saliminabi, M. Yildirim, R. W. Boyd, J-M. Ménard, and K. Dolgaleva, Optics Express 28, 3237-3248 (2020).

4. Suppression of self-focusing for few-cycle pulses, S. A. Kozlov, A. A. Drozdov, S. Choudhary, M. A. Kniazev, and R. W. Boyd, Journal of the Optical Society of America B 36, G68-G77 (2019).

### Quantum Communication

We have studied the utility of making use of bases that are not unbiased [1]. The utility of this study is that it has been commonly believed that an efficient QKD system must use two or more mutually unbiased bases. We have studied the possibility of intentionally using states spaces that are not mutually unbiased, and we have found that this situation can be helpful in mitigating the problem of state-dependent diffraction and also in simplifying some of the laboratory setup for producing qudit states and for measuring them.

1. High-dimensional quantum key distribution based on mutually partially unbiased bases, F. Wang, P. Zeng, J. Zhao, B. Braverman, Y. Zhou, M. Mirhosseini, X. Wang, H. Gao, F. Li, R. W. Boyd, and P. Zhang, Phys. Rev. A 101, 032340 (2020).

### Quantum Technologies

Canada is well positioned to become a world leader in studies of quantum phenomena and developing technologies based on quantum effects. Our research group has a major effort in exploring quantum effects both as a matter of pure science and as a means of developing quantum-enabled applications. During the past year we made significant progress towards these goals. We have performed a careful theoretical study of the limits imposed by quantum effects on the accuracy of measurements in ellipsometry [1]. We have also performed a laboratory investigation on the ability to perform nonlocal aberration correction through use of entangled photons [2]. In the study of fundamental aspects of quantum optics, we have explored a variation of the well-known Einstein-Podolsky-Rosen effect for photons entangled in radial position and radial momentum, a degree of freedom not previously studied [3]. We also performed an experiment in which we were able to make a quantum-limited estimation of the axial separation of two incoherent point sources [4].

- Fundamental quantum limits in ellipsometry,Ł. Rudnicki, L. L. Sánchez-Soto, G. Leuchs, and R. W. Boyd, Optics Letters 45, 4607-4610 (2020).
- Quantum Nonlocal Aberration Cancellation, A. N. Black, E. Giese, B. Braverman, N. Zollo, S. M. Barnett, and R. W. Boyd, Phys. Rev. Lett. 123, 143603 (2019).
- Realization of the Einstein-Podolsky-Rosen Paradox Using Radial Position and Radial Momentum Variables, L. Chen, T. Ma, X. Qiu, D. Zhang, W. Zhang, and R. W. Boyd, Phys. Rev. Lett. 123, 060403 (2019).
- Quantum-limited estimation of the axial separation of two incoherent point sources, Y. Zhou, J. Yang, J. D. Hassett, S. M. H. Rafsanjani, M. Mirhosseini, A. N. Vamivakas, A. N. Jordan, Z. Shi, and R. W. Boyd, Optica 6, 534-541 (2019).